Research ArticleModeling and Dynamic Analysis of Cutterhead DrivingSystem in Tunnel Boring Machine

Wei Sun Honghui Ma Xueguan Song Lintao Wang and Xin Ding

School of Mechanical Engineering Dalian University of Technology Dalian 116024 China

Correspondence should be addressed to Lintao Wang wltdluteducn

Received 12 June 2016 Accepted 29 November 2016 Published 9 January 2017

Academic Editor Evgeny Petrov

Copyright copy 2017 Wei Sun et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Failure of cutterhead driving system (CDS) of tunnel boring machine (TBM) often occurs under shock and vibration conditionsTo investigate the dynamic characteristics and reduce system vibration further an electromechanical coupling model of CDS isestablished which includes the model of direct torque control (DTC) system for three-phase asynchronous motor and purelytorsional dynamic model of multistage gear transmission system The proposed DTC model can provide driving torque just as thepractical inverter motor operates so that the influence of motor operating behavior will not be erroneously estimated Moreovernonlinear gear meshing factors such as time-variant mesh stiffness and transmission error are involved in the dynamic modelBased on the established nonlinear model of CDS vibration modes can be classified into three types that is rigid motion moderotational vibration mode and planet vibration mode Moreover dynamic responses under actual driving torque and idealizedequivalent torque are comparedwhich reveals that the ripple of actual driving torquewould aggravate vibration of gear transmissionsystem Influence index of torque ripple is proposed to show that vibration of system increases with torque ripple This studyprovides useful guideline for antivibration design and motor control of CDS in TBM

1 Introduction

Tunnel boring machine (TBM) is a large and high-techconstruction equipment which is widely used in transportmunicipal and water diversion projects due to its advantagessuch as highly integrated functions high tunneling speedand environment-friendly construction [1] Cutterhead is akey component of TBM which can crush and cut rock withdisc cutters mounted on its panel As shown in Figure 1cutterhead driving system (CDS) is a complex electrome-chanical coupling system which is mainly composed ofinverter motor planetary reducer pinion and ring gear

During tunneling process driving torque provided bymultiple inverter motor is transported from planetary re-ducer and pinions to ring gear which is fixedly connectedwith cutterhead Under the enlarged driving torque cutter-head rotates and breaks rock In CDS variable frequencyspeed control system such as vector control (VC) and direct

torque control (DTC) system are often applied to makeinverter motor respond quickly to the variable load

Due to complex and changing geological conditionscutterhead and its driving system often suffer large impactload with drastic fluctuation during tunnel construction [2]In addition the multistage gear transmission system is a timevarying and strong coupling system which causes periodicinternal excitation owning to the nonlinear factors such astime-variant mesh stiffness transmission error and backlash[3] Under such internal excitation formed in gear trans-mission system and external excitation caused by geologicalcondition and inverter motor server vibration in CDS oftenoccurs and results in failures such as excessive wear breakageof gear tooth and shaft imbalance of driving torque and soon [4] To resolve these problems and improve the design ofCDS the dynamic characteristics of gear transmission systemand the operating characters of inverter motor ought to beinvestigated primarily and imperatively

HindawiShock and VibrationVolume 2017 Article ID 7156816 12 pageshttpsdoiorg10115520177156816

2 Shock and Vibration

Cutter head

Cutter

Bearing

Ring

Pinion

Reducer

Motor

Driving system

Figure 1 Cutterhead driving system in TBM

In recent years a large amount of work has been doneon load sharing and vibration reduction of CDS in TBMNumerous researches have focused on dynamic analysis ofCDS Wei et al established a dynamic model of multigeardriving system and studied the effects of inertia on load-sharing characteristic [5 6] Sun et al established a dynamicmodel of cutterhead driving system based on hierarchicalmodelingmethod and obtained dynamic response [7] Zhanget al analyzed dynamic characteristic of TBM in mixed-face conditions [8] Qin and Zhao built multiobjective opti-mization model based on dynamic analysis and optimizedparameters of gear transmission system to reduce vibration[9] Besides multimotor synchronization control method ofCDS also attracts more andmore attentions Liu et al studiedload-sharing characteristic of multiple motors and proposedan adaptable control approach to improve the complianceability of CDS [10ndash12] All these researches havemade fruitfulefforts on design of CDS through dynamic analysis andoptimization of multimotor control strategy However thesestudies did not consider the influence of external excitationprovided by inverter motor which just replaced the actualdriving torque with an idealized constant value A number ofresearches have shown that torque ripple caused by variablefrequency speed control system is an inevitable factor whichmay influence the dynamic performance of transmissionmechanism [13] Without considering the ripple of actualdriving torque dynamic analysis of gear transmissionmay beerroneously investigatedTherefore it needs to take operatingcharacters of inverter motor into account for building theelectromechanical couplingmodel and studying the dynamiccharacteristic of multistage gear transmission system in CDS

In this paper dynamic analysis of CDS in TBM isstudied to explore the failure reasons of key componentsAn electromechanical coupling model of CDS is establishedwhich includes a dynamic model of DTC driving systemand a purely torsional dynamic model of multistage geartransmission system By taking the nonlinear factors of gearmeshing and the operating characters of inverter motor intoaccount dynamic characteristics of multistage gear transmis-sion system under the actual driving torque are analyzed It

provides data support for gear antivibration design andmotorcontrol of CDS in TBM

2 Mathematical Modeling of TBMCutterhead Driving System

21 Dynamic Model of DTC Driving System DTC system isparticularly applied to CDS with large inertia which needsrapid torque response Based on Bang-Bang control methodDTC system regulates stator flux and provides heavy startingtorque for CDS

In DTC driving system 120572-120573 phase static coordinatesystem is chosen as the reference frame of mathematicalmodel of three-phase asynchronous motor and hence thevoltage equation can be expressed as follows

[[[[[[

11990611990412057211990611990412057300

]]]]]]

= [[[[[[

119877119904 + 119901119871 119904 0 119901119871119898 00 119877119904 + 119901119871 119904 0 119901119871119898119901119871119898 119871 119904120596 119877119903 + 119901119871119903 119871119903120596minus120596119871119898 119901119871119898 minus120596119871119903 119877119903 + 119901119871119903

]]]]]]

[[[[[[

119894119904120572119894119904120573119894119903120572119894119903120573

]]]]]]

(1)

Flux Equation

[[[[[[

120595119904120572120595119904120573120595119903120572120595119903120573

]]]]]]

= [[[[[[

119871 119904 0 119871119898 00 119871 119904 0 119871119898119871119898 0 119871119903 00 119871119898 0 119871119903

]]]]]]

[[[[[[

119894119904120572119894119904120573119894119903120572119894119903120573

]]]]]]

(2)

Torque Equation

119879119890 = 119899119901119871119898 (119894119904120573119894119903120572 minus 119894119903120573119894119904120572) = 119899119901 (119894119904120573120595119904120572 minus 119894119904120572120595119904120573)= 119899119901 (120595119904 otimes 119894119904) (3)

where 119906119904120572 and 119906119904120573 are stator voltages 119894119904120572 119894119904120573 119894119903120572 and 119894119903120573 arestatorrotor currents 120595119904120572 120595119904120573 120595119903120572 and 120595119903120573 are statorrotorfluxes 119877119904 and 119877119903 are statorrotor resistances 119877119904 and 119877119903 arestatorrotor resistances 119871 119904 119871119903 and 119871119898 are statorrotor induc-tance and mutual inductance 119879119890 is electromagnet torque 120596is electrical angular speed of rotor 119899119901 is the number of polepairs and 119901 is differential operator

On the basis of (1)ndash(3) DTC system of CDS is establishedby Simulink module inMatlab software as shown in Figure 2The u-imodel is chosen as the stator flux observer which canbe expressed as follows

120595119904120572 = int (119906119904120572 minus 119877119904119894119904120572) 119889119905120595119904120573 = int (119906119904120573 minus 119877119904119894119904120573) 119889119905

(4)

Shock and Vibration 3

Continuous

powergui

1360

nlowast

+

+

minus

+

minus

+minusn

ASR

2

n

To workspace

Switch signal

Selector

XY graph

Gain

Universal bridge

DC

+minus

g

A

B

C

Observation

v

v

I_3s2s

U_3s2s

A

B

C

m

Asynchronous

SI units

-K-

machine

TLTm

n1 Tlowaste Tlowaste

Fluxlowasts

Fluxlowasts

Te1

Fluxsa

Fluxsa

Fluxsb

Fluxsb

V3 Uab

V1

Usa

Usb

Isa

Isb

Usalpha

Usbeta

Isalpha

Isbeta

Te

Te

Te

Te

Is_abc

Is_ab

Uab_Ubc

Figure 2 Direct torque control system of three-phase asynchronous motor

1

2

3

4

+minus

-K-

1

s

1

s

+minus

-K-

Integrator

Integrator 1

1

times

P

times

+minus 2

2

3

Rs Rs1

Te

Fluxsa

Fluxsb

Usa

Usb

Isa

Isb

PN

P1

Figure 3 Torque and stator flux observer model

According to (3)-(4) torque and stator flux observermodel is established as shown in Figure 3 In DTC systemthe amplitude of stator flux 120595119904 is kept constant and the angleof stator flux 120595119904 is regulated to control the electromagnettorque as shown in Figure 4 The asynchronous motor iscontrolled by switch status of voltage space vector in inverterDriving signals are selected from the optimal switching tableafter directly calculating stator flux and torque The locationof stator flux in 120572-120573 phase static coordinate system can becalculated by comparing the observed values of120595119904 and119879119890withthe given value of 120595lowast119904 and 119879lowast119890

Based on the model of DTC driving system frequencycontrol process of inverter motor can be simulated and elec-tromagnet torque 119879119890 can be obtained to drive the multistagegear transmission system

22 Dynamic Model of Multistage Gear Transmission SystemAs shown in Figure 5 multistage gear transmission system

120573

o

120595s

120572

120595lowasts

I

IIIII

IV

V VI

u1

u2u3

u4

u5 u6

Figure 4 Control principle of DTC system

is composed of three-stage planetary reducer and one-stagepinion-ring gears 119904(119894) 119903(119894) 119888(119894) and 119901(119894)119895 (119894 = 1 2 3 119895 =1 2 3 4) represent the ith-stage sun gear ring gear planetcarrier and the ith-stage jth planet gear in planetary reducer1198921 and 1198922 represent pinion-ring gears

Based on the lumped mass method a purely torsionaldynamic model of multistage gear transmission system isestablished Each component is regarded as a rigid body Thedirection of displacement along the meshing line is supposedto be positive when the tooth surface is under pressure BasedonNewtonrsquos SecondLaw the equivalentmathematicmodel of

4 Shock and Vibration

Tin

120579p

120579p

120579p

120579s

c1

k1

c2

k2

c3

k3 egkg

cg

Tout

g1

g2

es

ks

cs

120579s

es

ks

cs

120579p

erkr

cr120579c

er

kr

cr

s(i)

r(i)

c(i)

p(i)j

Figure 5 Purely torsional dynamic model of multistage gear transmission system

the multistage gear transmission system can be expressed asfollows

119868(1)119904 (1)119904 = 119879in minus 3sum119895=1

119896(1)119904119895 119909(1)119904119895 119903(1)119904 minus 3sum119895=1

119888(1)119904119895 (1)119904119895 119903(1)119904119868(1)1199011 (1)1199011 = 119896(1)1199041 119909(1)1199041 119903(1)1199011 minus 119896(1)1199031 119909(1)1199031 119903(1)1199011 + 119888(1)1199041 (1)1199041 119903(1)1199011

minus 119888(1)1199031 (1)1199031 119903(1)1199011119868(1)1199012 (1)1199012 = 119896(1)1199042 119909(1)1199042 119903(1)1199012 minus 119896(1)1199032 119909(1)1199032 119903(1)1199012 + 119888(1)1199042 (1)1199042 119903(1)1199012

minus 119888(1)1199032 (1)1199032 119903(1)1199012119868(1)1199013 (1)1199013 = 119896(1)1199043 119909(1)1199043 119903(1)1199013 minus 119896(1)1199033 119909(1)1199033 119903(1)1199013 + 119888(1)1199043 (1)1199043 119903(1)1199013

minus 119888(1)1199033 (1)1199033 119903(1)1199013119868(1)119888 (1)119888 = 3sum

119895=1

[(119896(1)119904119895 119909(1)119904119895 + 119896(1)119903119895 119909(1)119903119895 ) 119903(1)119888 cos120572]

+ 3sum119895=1

[(119888(1)119904119895 (1)119904119895 + 119888(1)119903119895 (1)119903119895 ) 119903(1)119888 cos120572] minus 119896(1)119888 120579(1)119888minus 119888(1)119888 (1)119888 minus 1198961 (120579(1)119888 minus 120579(2)119904 ) minus 1198881 ((1)119888 minus (2)119904 )

119868(2)119904 (2)119904 = 1198961 (120579(1)119888 minus 120579(2)119904 ) + 1198881 ((1)119888 minus (2)119904 )minus 4sum119895=1

119896(2)119904119895 119909(2)119904119895 119903(2)119904 minus 4sum119895=1

119888(2)119904119895 (2)119904119895 119903(2)119904119868(2)1199011 (2)1199011 = 119896(2)1199041 119909(2)1199041 119903(2)1199011 minus 119896(2)1199031 119909(2)1199031 119903(2)1199011 + 119888(2)1199041 (2)1199041 119903(2)1199011

minus 119888(2)1199031 (2)1199031 119903(2)1199011

119868(2)1199012 (2)1199012 = 119896(2)1199042 119909(2)1199042 119903(2)1199012 minus 119896(2)1199032 119909(2)1199032 119903(2)1199012 + 119888(2)1199042 (2)1199042 119903(2)1199012minus 119888(2)1199032 (2)1199032 119903(2)1199012

119868(2)1199013 (2)1199013 = 119896(2)1199043 119909(2)1199043 119903(2)1199013 minus 119896(2)1199033 119909(2)1199033 119903(2)1199013 + 119888(2)1199043 (2)1199043 119903(2)1199013minus 119888(2)1199033 (2)1199033 119903(2)1199013

119868(2)1199014 (2)1199014 = 119896(2)1199044 119909(2)1199044 119903(2)1199014 minus 119896(2)1199034 119909(2)1199034 119903(2)1199014 + 119888(2)1199044 (2)1199044 119903(2)1199014minus 119888(2)1199034 (2)1199034 119903(2)1199014

119868(2)119888 (2)119888 = 4sum119895=1

[(119896(2)119904119895 119909(2)119904119895 + 119896(2)119903119895 119909(2)119903119895 ) 119903(2)119888 cos120572]

+ 4sum119895=1

[(119888(2)119904119895 (2)119904119895 + 119888(2)119903119895 (2)119903119895 ) 119903(2)119888 cos120572] minus 119896(2)119888 120579(2)119888minus 119888(2)119888 (2)119888 minus 1198962 (120579(2)119888 minus 120579(3)119904 ) minus 1198882 ((2)119888 minus (3)119904 )

119868(3)119904 (3)119904 = 1198962 (120579(2)119888 minus 120579(3)119904 ) + 1198882 ((2)119888 minus (3)119904 )minus 4sum119895=1

119896(3)119904119895 119909(3)119904119895 119903(3)119904 minus 4sum119895=1

119888(3)119904119895 (3)119904119895 119903(3)119904119868(3)1199011 (3)1199011 = 119896(3)1199041 119909(3)1199041 119903(3)1199011 minus 119896(3)1199031 119909(3)1199031 119903(3)1199011 + 119888(3)1199041 (3)1199041 119903(3)1199011

minus 119888(3)1199031 (3)1199031 119903(3)1199011119868(3)1199012 (3)1199012 = 119896(3)1199042 119909(3)1199042 119903(3)1199012 minus 119896(3)1199032 119909(3)1199032 119903(3)1199012 + 119888(3)1199042 (3)1199042 119903(3)1199012

minus 119888(3)1199032 (3)1199032 119903(3)1199012

Shock and Vibration 5

119868(3)1199013 (3)1199013 = 119896(3)1199043 119909(3)1199043 119903(3)1199013 minus 119896(3)1199033 119909(3)1199033 119903(3)1199013 + 119888(3)1199043 (3)1199043 119903(3)1199013minus 119888(3)1199033 (3)1199033 119903(3)1199013

119868(3)1199014 (3)1199014 = 119896(3)1199044 119909(3)1199044 119903(3)1199014 minus 119896(3)1199034 119909(3)1199034 119903(3)1199014 + 119888(3)1199044 (3)1199044 119903(3)1199014minus 119888(3)1199034 (3)1199034 119903(3)1199014

119868(3)119888 (3)119888 = 4sum119895=1

[(119896(3)119904119895 119909(3)119904119895 + 119896(3)119903119895 119909(3)119903119895 ) 119903(3)119888 cos120572]

+ 4sum119895=1

[(119888(3)119904119895 (3)119904119895 + 119888(3)119903119895 (3)119903119895 ) 119903(3)119888 cos120572] minus 119896(3)119888 120579(3)119888minus 119888(3)119888 (3)119888 minus 1198963 (120579(3)119888 minus 1205791198921) minus 1198883 ((3)119888 minus 1198921)

11986811989211198921 = 1198963 (120579(3)119888 minus 1205791198921) + 1198883 ((3)119888 minus 1198921) minus 119896119892 (11990311989211205791198921minus 11990311989221205791198922 + 119890119892) minus 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)

11986811989221198922 = 119899 [119896119892 (11990311989211205791198921 minus 11990311989221205791198922 + 119890119892)+ 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)] minus 119879out

(5)

where 119868119904 119868119901 119868119888 1198681198921 and 1198681198922 are mass moments of inertia ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119903119904 119903119901 119903119888 1199031198921 and 1199031198922 are base radiuses of sungear planet gear planet carrier in reducer and pinion-ringgears 120579119904 120579119901 120579119888 1205791198921 and 1205791198922 are angular displacements ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119879in is driving torque of inverter motor which isequal to electromagnet torque 119879119890 in DTC system 119879out is theenlarged driving torque by gear transmission system 119896119888 istorsional stiffness of planet carrier 1198961 1198962 and 1198963 are torsionalstiffnesses of each stage connecting stage 119888119888 is torsionaldamping of planet carrier 1198881 1198882 and 1198883 are torsional dampingsof each stage connecting stage 120572 is pressure angle at the pitchcylinder 119899 is number of pinions 119909119904 is displacement along themeshing line between the sun gear and each planet gear and119909119903 is displacement along the meshing line between the ringgear and each planet gear119909119904 and 119909119903 can be expressed as follows

119909(119894)119904119895 = 119903(119894)119904 120579(119894)119904 minus 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119904119895119909(119894)119903119895 = 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119903119895

(119894 = 1 2 3 119895 = 1 2 3 4) (6)

where 119890119904 is transmission error between the sun gear and eachplanet gear and 119890119903 is transmission error between the ring gearand each planet gear

0 001 002 003 004 005 006 007 008 00913

14

15

16

17

18

19

2

21

Mes

h sti

ffnes

s (N

m)

Time (s)

times109

120596m = 167

km = 175 times 109

Figure 6 Time-varying mesh stiffness

0 05 1 15 2 25 3minus2

minus15

minus1

minus05

0

05

1

15

2

Time (s)

Mes

h er

ror (

m)

times10minus5

120596m = 647

120596s = 94

Fp = 317 times 10minus5

f998400p = 171 times 10minus5

Figure 7 Transmission mesh error

As shown in Figure 6 119896119904 119896119903 and 119896119892 are time-variantmeshstiffnesses which can be expressed by means of the Fourierseries expansion as follows [14]

119896119898 (119905) = 119896119898 + 119873sum119899=1

119861119899 cos 119894120596119898 (119905 + 120593) 119898 = 119904 119903 119892 (7)

where 119896119898 is average mesh stiffness which can be obtainedbased on gear standards such as AGMA ISO 1328-1 andDIN3990 and119861119899 is the n-rank harmonic amplitude in Fourierseries119888119904 119888119903 and 119888119892 are mesh dampings which can be expressedas follows

119888119898 = 2120589radic 119896119898119898119898119898119899119898119898 + 119898119899 119898 or 119899 = 119904 119903 119901 119892 (8)

where 120589 is gear mesh damping ratio (120589 = 003ndash017) and 119898119898and119898119899 are masses of two meshing gears

As shown in Figure 7 transmission error 119890119899 is approx-imated as superposition of harmonic function of meshfrequency and rotation frequency of shaft [15]

119890119899 = 05119865119901 sin (2120587120596119904119905 + 120593119904) + 051198911015840119901 sin (2120587120596119898119905 + 120593119898)119899 = 119904 119903 119892 (9)

6 Shock and Vibration

Table 1 Technical parameters of TBM cutterhead driving system

Driving motor Rated power 160 kWSpeed range 0ndash1480 rpm

Transmission system Reducer Gear ratio 119894I = 512Ring-pinion Gear ratio 119894II = 126

CutterheadRated power 1600 kW (10lowast160 kW)Speed range 0ndash21 rpmndash47 rpmRated torque 7230KNm 21 rpm

Table 2 Parameters of three-phase asynchronous motor

Parameters ValueRated power 119875119873 160 kWRated voltage 119880119873 400VRated frequency 119891119873 50HzStator resistance 119877119904 001379ΩRotor resistance 119877119903 0007728ΩStator inductance 119871 119904 0152mHRotor inductance 119871 119903 0152mHMutual inductance 119871119898 769mHRotational inertia 119869 29 kgsdotm2

where 119865119901 is total cumulative pitch error 1198911015840119901 is tangentialtolerance of single tooth 120596119904 and 120596119898 are rotation frequencyand mesh frequency and 120593119904 and 120593119898 are initial phase of shaftand mesh phase

3 Dynamic Analysis of ElectromechanicalCoupling Model of CDS

31 Actual Driving Torque of DTC System The technicalparameters of one certain CDS are shown in Table 1According to these parameters the model of three-phaseasynchronous motor is chosen as Table 2 shows and thecontrol parameters of DTC system are set In this paper themultiple invertermotors are supposed to be synchronous andTBM cutterhead is chosen to work under the rotational speed119899119888 = 21 rpm Thus load torque of motor can be calculatedbased on the mean value of load torque on cutterhead whichcan be expressed as (10) shows

119879119871 = 9549 119875119873119894I119894II119899119888119899 (10)

where 119875119873 is rated power 119894I is gear ratio of reducer 119894II is gearratio of ring-pinion gears 119899119888 is rated speed of cutterhead and119899 is number of pinions

Field test data of external load torque is shown in Figure 9In actual tunneling process load torque 119879119871 is unstable andchanges abruptly as geological condition varies On the basisof (10) rated 119879119871 is 1120Nsdotm under rated rotational speed119899119888 = 21 rpm which corresponds to the actual 119879119871 near 310 sin Figure 8 Thus taking a 1 s-length (of) actual 119879119871 between3142 s and 3152 s as an example 119879119871 in the first 02 s keepsstable near rated torque and then rises sharply to 1700Nsdotm at3144 s After 3145 s119879119871 remains roughly stable near 1700Nsdotm

010

020

030

040

050

060

070

080

090

010

0011

0012

0013

0014

0015

00

0500

1000150020002500300035004000

3142 3144 3146 3148 31510001200140016001800

Time t (s)

Actu

al lo

ad to

rque

TL

(Nmiddotm

)

Figure 8 Field test data of external load torque

0 02 04 06 08 1 12 14 16 18 20

500

1000

1500

2000

2500

1 15 2

1000

1500

2000

DTC driving torqueActual driving torque

Time t (s)

Elec

trom

agne

t tor

queT

e(Nmiddotm

)

Figure 9 Actual driving torque of DTC system

with little fluctuations To study the operating charactersof inverter motor under shocking load the 1 s-length of 119879119871between 3142 s and 3152 s is chosen to be simulated as apiecewise function In DTC driving system load torque 119879119871is simulated for 2 s 119879119871 is set to be 1100Nsdotm before 135 s and119879119871 is equal to 1700Nsdotm during 135 s and 2 s

The actual driving torque of DTC system is obtained andshown in Figure 9 In the start-up phase inverter motoroperates with the maximum torque to accelerate to therated speed quickly After operating for 1 s electromagnetictorque 119879119890 fits the actual load torque under rated speedThe fitting result shows that DTC driving system respondsquickly according to the changing load torque 119879119871 Howeverelectromagnetic torque 119879119890 has high torque ripple which isabout 120Nsdotm which can be expressed in discrete form asfollows [16]

119879(119896+1)119890 = 119879(119896)119890 + Δ119879(119896)1198901 + Δ119879(119896)1198901Δ119879(119896)1198901 = minus119879(119896)119890 (119877119904119871 119904 +

119877119903119871119903)119879119904120590

Δ119879(119896)1198902 = 32119899119901 119871119898120590119871 119904119871119903 [(119906(119896)119904 minus 119895120596(119896)119903 120595(119896)119904 ) sdot 119895120595(119896)119903 ] 119879119904(11)

Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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International Journal of

Shock and Vibration 3

Continuous

powergui

1360

nlowast

+

+

minus

+

minus

+minusn

ASR

2

n

To workspace

Switch signal

Selector

XY graph

Gain

Universal bridge

DC

+minus

g

A

B

C

Observation

v

v

I_3s2s

U_3s2s

A

B

C

m

Asynchronous

SI units

-K-

machine

TLTm

n1 Tlowaste Tlowaste

Fluxlowasts

Fluxlowasts

Te1

Fluxsa

Fluxsa

Fluxsb

Fluxsb

V3 Uab

V1

Usa

Usb

Isa

Isb

Usalpha

Usbeta

Isalpha

Isbeta

Te

Te

Te

Te

Is_abc

Is_ab

Uab_Ubc

Figure 2 Direct torque control system of three-phase asynchronous motor

1

2

3

4

+minus

-K-

1

s

1

s

+minus

-K-

Integrator

Integrator 1

1

times

P

times

+minus 2

2

3

Rs Rs1

Te

Fluxsa

Fluxsb

Usa

Usb

Isa

Isb

PN

P1

Figure 3 Torque and stator flux observer model

According to (3)-(4) torque and stator flux observermodel is established as shown in Figure 3 In DTC systemthe amplitude of stator flux 120595119904 is kept constant and the angleof stator flux 120595119904 is regulated to control the electromagnettorque as shown in Figure 4 The asynchronous motor iscontrolled by switch status of voltage space vector in inverterDriving signals are selected from the optimal switching tableafter directly calculating stator flux and torque The locationof stator flux in 120572-120573 phase static coordinate system can becalculated by comparing the observed values of120595119904 and119879119890withthe given value of 120595lowast119904 and 119879lowast119890

Based on the model of DTC driving system frequencycontrol process of inverter motor can be simulated and elec-tromagnet torque 119879119890 can be obtained to drive the multistagegear transmission system

22 Dynamic Model of Multistage Gear Transmission SystemAs shown in Figure 5 multistage gear transmission system

120573

o

120595s

120572

120595lowasts

I

IIIII

IV

V VI

u1

u2u3

u4

u5 u6

Figure 4 Control principle of DTC system

is composed of three-stage planetary reducer and one-stagepinion-ring gears 119904(119894) 119903(119894) 119888(119894) and 119901(119894)119895 (119894 = 1 2 3 119895 =1 2 3 4) represent the ith-stage sun gear ring gear planetcarrier and the ith-stage jth planet gear in planetary reducer1198921 and 1198922 represent pinion-ring gears

Based on the lumped mass method a purely torsionaldynamic model of multistage gear transmission system isestablished Each component is regarded as a rigid body Thedirection of displacement along the meshing line is supposedto be positive when the tooth surface is under pressure BasedonNewtonrsquos SecondLaw the equivalentmathematicmodel of

4 Shock and Vibration

Tin

120579p

120579p

120579p

120579s

c1

k1

c2

k2

c3

k3 egkg

cg

Tout

g1

g2

es

ks

cs

120579s

es

ks

cs

120579p

erkr

cr120579c

er

kr

cr

s(i)

r(i)

c(i)

p(i)j

Figure 5 Purely torsional dynamic model of multistage gear transmission system

the multistage gear transmission system can be expressed asfollows

119868(1)119904 (1)119904 = 119879in minus 3sum119895=1

119896(1)119904119895 119909(1)119904119895 119903(1)119904 minus 3sum119895=1

119888(1)119904119895 (1)119904119895 119903(1)119904119868(1)1199011 (1)1199011 = 119896(1)1199041 119909(1)1199041 119903(1)1199011 minus 119896(1)1199031 119909(1)1199031 119903(1)1199011 + 119888(1)1199041 (1)1199041 119903(1)1199011

minus 119888(1)1199031 (1)1199031 119903(1)1199011119868(1)1199012 (1)1199012 = 119896(1)1199042 119909(1)1199042 119903(1)1199012 minus 119896(1)1199032 119909(1)1199032 119903(1)1199012 + 119888(1)1199042 (1)1199042 119903(1)1199012

minus 119888(1)1199032 (1)1199032 119903(1)1199012119868(1)1199013 (1)1199013 = 119896(1)1199043 119909(1)1199043 119903(1)1199013 minus 119896(1)1199033 119909(1)1199033 119903(1)1199013 + 119888(1)1199043 (1)1199043 119903(1)1199013

minus 119888(1)1199033 (1)1199033 119903(1)1199013119868(1)119888 (1)119888 = 3sum

119895=1

[(119896(1)119904119895 119909(1)119904119895 + 119896(1)119903119895 119909(1)119903119895 ) 119903(1)119888 cos120572]

+ 3sum119895=1

[(119888(1)119904119895 (1)119904119895 + 119888(1)119903119895 (1)119903119895 ) 119903(1)119888 cos120572] minus 119896(1)119888 120579(1)119888minus 119888(1)119888 (1)119888 minus 1198961 (120579(1)119888 minus 120579(2)119904 ) minus 1198881 ((1)119888 minus (2)119904 )

119868(2)119904 (2)119904 = 1198961 (120579(1)119888 minus 120579(2)119904 ) + 1198881 ((1)119888 minus (2)119904 )minus 4sum119895=1

119896(2)119904119895 119909(2)119904119895 119903(2)119904 minus 4sum119895=1

119888(2)119904119895 (2)119904119895 119903(2)119904119868(2)1199011 (2)1199011 = 119896(2)1199041 119909(2)1199041 119903(2)1199011 minus 119896(2)1199031 119909(2)1199031 119903(2)1199011 + 119888(2)1199041 (2)1199041 119903(2)1199011

minus 119888(2)1199031 (2)1199031 119903(2)1199011

119868(2)1199012 (2)1199012 = 119896(2)1199042 119909(2)1199042 119903(2)1199012 minus 119896(2)1199032 119909(2)1199032 119903(2)1199012 + 119888(2)1199042 (2)1199042 119903(2)1199012minus 119888(2)1199032 (2)1199032 119903(2)1199012

119868(2)1199013 (2)1199013 = 119896(2)1199043 119909(2)1199043 119903(2)1199013 minus 119896(2)1199033 119909(2)1199033 119903(2)1199013 + 119888(2)1199043 (2)1199043 119903(2)1199013minus 119888(2)1199033 (2)1199033 119903(2)1199013

119868(2)1199014 (2)1199014 = 119896(2)1199044 119909(2)1199044 119903(2)1199014 minus 119896(2)1199034 119909(2)1199034 119903(2)1199014 + 119888(2)1199044 (2)1199044 119903(2)1199014minus 119888(2)1199034 (2)1199034 119903(2)1199014

119868(2)119888 (2)119888 = 4sum119895=1

[(119896(2)119904119895 119909(2)119904119895 + 119896(2)119903119895 119909(2)119903119895 ) 119903(2)119888 cos120572]

+ 4sum119895=1

[(119888(2)119904119895 (2)119904119895 + 119888(2)119903119895 (2)119903119895 ) 119903(2)119888 cos120572] minus 119896(2)119888 120579(2)119888minus 119888(2)119888 (2)119888 minus 1198962 (120579(2)119888 minus 120579(3)119904 ) minus 1198882 ((2)119888 minus (3)119904 )

119868(3)119904 (3)119904 = 1198962 (120579(2)119888 minus 120579(3)119904 ) + 1198882 ((2)119888 minus (3)119904 )minus 4sum119895=1

119896(3)119904119895 119909(3)119904119895 119903(3)119904 minus 4sum119895=1

119888(3)119904119895 (3)119904119895 119903(3)119904119868(3)1199011 (3)1199011 = 119896(3)1199041 119909(3)1199041 119903(3)1199011 minus 119896(3)1199031 119909(3)1199031 119903(3)1199011 + 119888(3)1199041 (3)1199041 119903(3)1199011

minus 119888(3)1199031 (3)1199031 119903(3)1199011119868(3)1199012 (3)1199012 = 119896(3)1199042 119909(3)1199042 119903(3)1199012 minus 119896(3)1199032 119909(3)1199032 119903(3)1199012 + 119888(3)1199042 (3)1199042 119903(3)1199012

minus 119888(3)1199032 (3)1199032 119903(3)1199012

Shock and Vibration 5

119868(3)1199013 (3)1199013 = 119896(3)1199043 119909(3)1199043 119903(3)1199013 minus 119896(3)1199033 119909(3)1199033 119903(3)1199013 + 119888(3)1199043 (3)1199043 119903(3)1199013minus 119888(3)1199033 (3)1199033 119903(3)1199013

119868(3)1199014 (3)1199014 = 119896(3)1199044 119909(3)1199044 119903(3)1199014 minus 119896(3)1199034 119909(3)1199034 119903(3)1199014 + 119888(3)1199044 (3)1199044 119903(3)1199014minus 119888(3)1199034 (3)1199034 119903(3)1199014

119868(3)119888 (3)119888 = 4sum119895=1

[(119896(3)119904119895 119909(3)119904119895 + 119896(3)119903119895 119909(3)119903119895 ) 119903(3)119888 cos120572]

+ 4sum119895=1

[(119888(3)119904119895 (3)119904119895 + 119888(3)119903119895 (3)119903119895 ) 119903(3)119888 cos120572] minus 119896(3)119888 120579(3)119888minus 119888(3)119888 (3)119888 minus 1198963 (120579(3)119888 minus 1205791198921) minus 1198883 ((3)119888 minus 1198921)

11986811989211198921 = 1198963 (120579(3)119888 minus 1205791198921) + 1198883 ((3)119888 minus 1198921) minus 119896119892 (11990311989211205791198921minus 11990311989221205791198922 + 119890119892) minus 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)

11986811989221198922 = 119899 [119896119892 (11990311989211205791198921 minus 11990311989221205791198922 + 119890119892)+ 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)] minus 119879out

(5)

where 119868119904 119868119901 119868119888 1198681198921 and 1198681198922 are mass moments of inertia ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119903119904 119903119901 119903119888 1199031198921 and 1199031198922 are base radiuses of sungear planet gear planet carrier in reducer and pinion-ringgears 120579119904 120579119901 120579119888 1205791198921 and 1205791198922 are angular displacements ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119879in is driving torque of inverter motor which isequal to electromagnet torque 119879119890 in DTC system 119879out is theenlarged driving torque by gear transmission system 119896119888 istorsional stiffness of planet carrier 1198961 1198962 and 1198963 are torsionalstiffnesses of each stage connecting stage 119888119888 is torsionaldamping of planet carrier 1198881 1198882 and 1198883 are torsional dampingsof each stage connecting stage 120572 is pressure angle at the pitchcylinder 119899 is number of pinions 119909119904 is displacement along themeshing line between the sun gear and each planet gear and119909119903 is displacement along the meshing line between the ringgear and each planet gear119909119904 and 119909119903 can be expressed as follows

119909(119894)119904119895 = 119903(119894)119904 120579(119894)119904 minus 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119904119895119909(119894)119903119895 = 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119903119895

(119894 = 1 2 3 119895 = 1 2 3 4) (6)

where 119890119904 is transmission error between the sun gear and eachplanet gear and 119890119903 is transmission error between the ring gearand each planet gear

0 001 002 003 004 005 006 007 008 00913

14

15

16

17

18

19

2

21

Mes

h sti

ffnes

s (N

m)

Time (s)

times109

120596m = 167

km = 175 times 109

Figure 6 Time-varying mesh stiffness

0 05 1 15 2 25 3minus2

minus15

minus1

minus05

0

05

1

15

2

Time (s)

Mes

h er

ror (

m)

times10minus5

120596m = 647

120596s = 94

Fp = 317 times 10minus5

f998400p = 171 times 10minus5

Figure 7 Transmission mesh error

As shown in Figure 6 119896119904 119896119903 and 119896119892 are time-variantmeshstiffnesses which can be expressed by means of the Fourierseries expansion as follows [14]

119896119898 (119905) = 119896119898 + 119873sum119899=1

119861119899 cos 119894120596119898 (119905 + 120593) 119898 = 119904 119903 119892 (7)

where 119896119898 is average mesh stiffness which can be obtainedbased on gear standards such as AGMA ISO 1328-1 andDIN3990 and119861119899 is the n-rank harmonic amplitude in Fourierseries119888119904 119888119903 and 119888119892 are mesh dampings which can be expressedas follows

119888119898 = 2120589radic 119896119898119898119898119898119899119898119898 + 119898119899 119898 or 119899 = 119904 119903 119901 119892 (8)

where 120589 is gear mesh damping ratio (120589 = 003ndash017) and 119898119898and119898119899 are masses of two meshing gears

As shown in Figure 7 transmission error 119890119899 is approx-imated as superposition of harmonic function of meshfrequency and rotation frequency of shaft [15]

119890119899 = 05119865119901 sin (2120587120596119904119905 + 120593119904) + 051198911015840119901 sin (2120587120596119898119905 + 120593119898)119899 = 119904 119903 119892 (9)

6 Shock and Vibration

Table 1 Technical parameters of TBM cutterhead driving system

Driving motor Rated power 160 kWSpeed range 0ndash1480 rpm

Transmission system Reducer Gear ratio 119894I = 512Ring-pinion Gear ratio 119894II = 126

CutterheadRated power 1600 kW (10lowast160 kW)Speed range 0ndash21 rpmndash47 rpmRated torque 7230KNm 21 rpm

Table 2 Parameters of three-phase asynchronous motor

Parameters ValueRated power 119875119873 160 kWRated voltage 119880119873 400VRated frequency 119891119873 50HzStator resistance 119877119904 001379ΩRotor resistance 119877119903 0007728ΩStator inductance 119871 119904 0152mHRotor inductance 119871 119903 0152mHMutual inductance 119871119898 769mHRotational inertia 119869 29 kgsdotm2

where 119865119901 is total cumulative pitch error 1198911015840119901 is tangentialtolerance of single tooth 120596119904 and 120596119898 are rotation frequencyand mesh frequency and 120593119904 and 120593119898 are initial phase of shaftand mesh phase

3 Dynamic Analysis of ElectromechanicalCoupling Model of CDS

31 Actual Driving Torque of DTC System The technicalparameters of one certain CDS are shown in Table 1According to these parameters the model of three-phaseasynchronous motor is chosen as Table 2 shows and thecontrol parameters of DTC system are set In this paper themultiple invertermotors are supposed to be synchronous andTBM cutterhead is chosen to work under the rotational speed119899119888 = 21 rpm Thus load torque of motor can be calculatedbased on the mean value of load torque on cutterhead whichcan be expressed as (10) shows

119879119871 = 9549 119875119873119894I119894II119899119888119899 (10)

where 119875119873 is rated power 119894I is gear ratio of reducer 119894II is gearratio of ring-pinion gears 119899119888 is rated speed of cutterhead and119899 is number of pinions

Field test data of external load torque is shown in Figure 9In actual tunneling process load torque 119879119871 is unstable andchanges abruptly as geological condition varies On the basisof (10) rated 119879119871 is 1120Nsdotm under rated rotational speed119899119888 = 21 rpm which corresponds to the actual 119879119871 near 310 sin Figure 8 Thus taking a 1 s-length (of) actual 119879119871 between3142 s and 3152 s as an example 119879119871 in the first 02 s keepsstable near rated torque and then rises sharply to 1700Nsdotm at3144 s After 3145 s119879119871 remains roughly stable near 1700Nsdotm

010

020

030

040

050

060

070

080

090

010

0011

0012

0013

0014

0015

00

0500

1000150020002500300035004000

3142 3144 3146 3148 31510001200140016001800

Time t (s)

Actu

al lo

ad to

rque

TL

(Nmiddotm

)

Figure 8 Field test data of external load torque

0 02 04 06 08 1 12 14 16 18 20

500

1000

1500

2000

2500

1 15 2

1000

1500

2000

DTC driving torqueActual driving torque

Time t (s)

Elec

trom

agne

t tor

queT

e(Nmiddotm

)

Figure 9 Actual driving torque of DTC system

with little fluctuations To study the operating charactersof inverter motor under shocking load the 1 s-length of 119879119871between 3142 s and 3152 s is chosen to be simulated as apiecewise function In DTC driving system load torque 119879119871is simulated for 2 s 119879119871 is set to be 1100Nsdotm before 135 s and119879119871 is equal to 1700Nsdotm during 135 s and 2 s

The actual driving torque of DTC system is obtained andshown in Figure 9 In the start-up phase inverter motoroperates with the maximum torque to accelerate to therated speed quickly After operating for 1 s electromagnetictorque 119879119890 fits the actual load torque under rated speedThe fitting result shows that DTC driving system respondsquickly according to the changing load torque 119879119871 Howeverelectromagnetic torque 119879119890 has high torque ripple which isabout 120Nsdotm which can be expressed in discrete form asfollows [16]

119879(119896+1)119890 = 119879(119896)119890 + Δ119879(119896)1198901 + Δ119879(119896)1198901Δ119879(119896)1198901 = minus119879(119896)119890 (119877119904119871 119904 +

119877119903119871119903)119879119904120590

Δ119879(119896)1198902 = 32119899119901 119871119898120590119871 119904119871119903 [(119906(119896)119904 minus 119895120596(119896)119903 120595(119896)119904 ) sdot 119895120595(119896)119903 ] 119879119904(11)

Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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Submit your manuscripts athttpswwwhindawicom

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Shock and Vibration

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

Tin

120579p

120579p

120579p

120579s

c1

k1

c2

k2

c3

k3 egkg

cg

Tout

g1

g2

es

ks

cs

120579s

es

ks

cs

120579p

erkr

cr120579c

er

kr

cr

s(i)

r(i)

c(i)

p(i)j

Figure 5 Purely torsional dynamic model of multistage gear transmission system

the multistage gear transmission system can be expressed asfollows

119868(1)119904 (1)119904 = 119879in minus 3sum119895=1

119896(1)119904119895 119909(1)119904119895 119903(1)119904 minus 3sum119895=1

119888(1)119904119895 (1)119904119895 119903(1)119904119868(1)1199011 (1)1199011 = 119896(1)1199041 119909(1)1199041 119903(1)1199011 minus 119896(1)1199031 119909(1)1199031 119903(1)1199011 + 119888(1)1199041 (1)1199041 119903(1)1199011

minus 119888(1)1199031 (1)1199031 119903(1)1199011119868(1)1199012 (1)1199012 = 119896(1)1199042 119909(1)1199042 119903(1)1199012 minus 119896(1)1199032 119909(1)1199032 119903(1)1199012 + 119888(1)1199042 (1)1199042 119903(1)1199012

minus 119888(1)1199032 (1)1199032 119903(1)1199012119868(1)1199013 (1)1199013 = 119896(1)1199043 119909(1)1199043 119903(1)1199013 minus 119896(1)1199033 119909(1)1199033 119903(1)1199013 + 119888(1)1199043 (1)1199043 119903(1)1199013

minus 119888(1)1199033 (1)1199033 119903(1)1199013119868(1)119888 (1)119888 = 3sum

119895=1

[(119896(1)119904119895 119909(1)119904119895 + 119896(1)119903119895 119909(1)119903119895 ) 119903(1)119888 cos120572]

+ 3sum119895=1

[(119888(1)119904119895 (1)119904119895 + 119888(1)119903119895 (1)119903119895 ) 119903(1)119888 cos120572] minus 119896(1)119888 120579(1)119888minus 119888(1)119888 (1)119888 minus 1198961 (120579(1)119888 minus 120579(2)119904 ) minus 1198881 ((1)119888 minus (2)119904 )

119868(2)119904 (2)119904 = 1198961 (120579(1)119888 minus 120579(2)119904 ) + 1198881 ((1)119888 minus (2)119904 )minus 4sum119895=1

119896(2)119904119895 119909(2)119904119895 119903(2)119904 minus 4sum119895=1

119888(2)119904119895 (2)119904119895 119903(2)119904119868(2)1199011 (2)1199011 = 119896(2)1199041 119909(2)1199041 119903(2)1199011 minus 119896(2)1199031 119909(2)1199031 119903(2)1199011 + 119888(2)1199041 (2)1199041 119903(2)1199011

minus 119888(2)1199031 (2)1199031 119903(2)1199011

119868(2)1199012 (2)1199012 = 119896(2)1199042 119909(2)1199042 119903(2)1199012 minus 119896(2)1199032 119909(2)1199032 119903(2)1199012 + 119888(2)1199042 (2)1199042 119903(2)1199012minus 119888(2)1199032 (2)1199032 119903(2)1199012

119868(2)1199013 (2)1199013 = 119896(2)1199043 119909(2)1199043 119903(2)1199013 minus 119896(2)1199033 119909(2)1199033 119903(2)1199013 + 119888(2)1199043 (2)1199043 119903(2)1199013minus 119888(2)1199033 (2)1199033 119903(2)1199013

119868(2)1199014 (2)1199014 = 119896(2)1199044 119909(2)1199044 119903(2)1199014 minus 119896(2)1199034 119909(2)1199034 119903(2)1199014 + 119888(2)1199044 (2)1199044 119903(2)1199014minus 119888(2)1199034 (2)1199034 119903(2)1199014

119868(2)119888 (2)119888 = 4sum119895=1

[(119896(2)119904119895 119909(2)119904119895 + 119896(2)119903119895 119909(2)119903119895 ) 119903(2)119888 cos120572]

+ 4sum119895=1

[(119888(2)119904119895 (2)119904119895 + 119888(2)119903119895 (2)119903119895 ) 119903(2)119888 cos120572] minus 119896(2)119888 120579(2)119888minus 119888(2)119888 (2)119888 minus 1198962 (120579(2)119888 minus 120579(3)119904 ) minus 1198882 ((2)119888 minus (3)119904 )

119868(3)119904 (3)119904 = 1198962 (120579(2)119888 minus 120579(3)119904 ) + 1198882 ((2)119888 minus (3)119904 )minus 4sum119895=1

119896(3)119904119895 119909(3)119904119895 119903(3)119904 minus 4sum119895=1

119888(3)119904119895 (3)119904119895 119903(3)119904119868(3)1199011 (3)1199011 = 119896(3)1199041 119909(3)1199041 119903(3)1199011 minus 119896(3)1199031 119909(3)1199031 119903(3)1199011 + 119888(3)1199041 (3)1199041 119903(3)1199011

minus 119888(3)1199031 (3)1199031 119903(3)1199011119868(3)1199012 (3)1199012 = 119896(3)1199042 119909(3)1199042 119903(3)1199012 minus 119896(3)1199032 119909(3)1199032 119903(3)1199012 + 119888(3)1199042 (3)1199042 119903(3)1199012

minus 119888(3)1199032 (3)1199032 119903(3)1199012

Shock and Vibration 5

119868(3)1199013 (3)1199013 = 119896(3)1199043 119909(3)1199043 119903(3)1199013 minus 119896(3)1199033 119909(3)1199033 119903(3)1199013 + 119888(3)1199043 (3)1199043 119903(3)1199013minus 119888(3)1199033 (3)1199033 119903(3)1199013

119868(3)1199014 (3)1199014 = 119896(3)1199044 119909(3)1199044 119903(3)1199014 minus 119896(3)1199034 119909(3)1199034 119903(3)1199014 + 119888(3)1199044 (3)1199044 119903(3)1199014minus 119888(3)1199034 (3)1199034 119903(3)1199014

119868(3)119888 (3)119888 = 4sum119895=1

[(119896(3)119904119895 119909(3)119904119895 + 119896(3)119903119895 119909(3)119903119895 ) 119903(3)119888 cos120572]

+ 4sum119895=1

[(119888(3)119904119895 (3)119904119895 + 119888(3)119903119895 (3)119903119895 ) 119903(3)119888 cos120572] minus 119896(3)119888 120579(3)119888minus 119888(3)119888 (3)119888 minus 1198963 (120579(3)119888 minus 1205791198921) minus 1198883 ((3)119888 minus 1198921)

11986811989211198921 = 1198963 (120579(3)119888 minus 1205791198921) + 1198883 ((3)119888 minus 1198921) minus 119896119892 (11990311989211205791198921minus 11990311989221205791198922 + 119890119892) minus 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)

11986811989221198922 = 119899 [119896119892 (11990311989211205791198921 minus 11990311989221205791198922 + 119890119892)+ 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)] minus 119879out

(5)

where 119868119904 119868119901 119868119888 1198681198921 and 1198681198922 are mass moments of inertia ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119903119904 119903119901 119903119888 1199031198921 and 1199031198922 are base radiuses of sungear planet gear planet carrier in reducer and pinion-ringgears 120579119904 120579119901 120579119888 1205791198921 and 1205791198922 are angular displacements ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119879in is driving torque of inverter motor which isequal to electromagnet torque 119879119890 in DTC system 119879out is theenlarged driving torque by gear transmission system 119896119888 istorsional stiffness of planet carrier 1198961 1198962 and 1198963 are torsionalstiffnesses of each stage connecting stage 119888119888 is torsionaldamping of planet carrier 1198881 1198882 and 1198883 are torsional dampingsof each stage connecting stage 120572 is pressure angle at the pitchcylinder 119899 is number of pinions 119909119904 is displacement along themeshing line between the sun gear and each planet gear and119909119903 is displacement along the meshing line between the ringgear and each planet gear119909119904 and 119909119903 can be expressed as follows

119909(119894)119904119895 = 119903(119894)119904 120579(119894)119904 minus 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119904119895119909(119894)119903119895 = 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119903119895

(119894 = 1 2 3 119895 = 1 2 3 4) (6)

where 119890119904 is transmission error between the sun gear and eachplanet gear and 119890119903 is transmission error between the ring gearand each planet gear

0 001 002 003 004 005 006 007 008 00913

14

15

16

17

18

19

2

21

Mes

h sti

ffnes

s (N

m)

Time (s)

times109

120596m = 167

km = 175 times 109

Figure 6 Time-varying mesh stiffness

0 05 1 15 2 25 3minus2

minus15

minus1

minus05

0

05

1

15

2

Time (s)

Mes

h er

ror (

m)

times10minus5

120596m = 647

120596s = 94

Fp = 317 times 10minus5

f998400p = 171 times 10minus5

Figure 7 Transmission mesh error

As shown in Figure 6 119896119904 119896119903 and 119896119892 are time-variantmeshstiffnesses which can be expressed by means of the Fourierseries expansion as follows [14]

119896119898 (119905) = 119896119898 + 119873sum119899=1

119861119899 cos 119894120596119898 (119905 + 120593) 119898 = 119904 119903 119892 (7)

where 119896119898 is average mesh stiffness which can be obtainedbased on gear standards such as AGMA ISO 1328-1 andDIN3990 and119861119899 is the n-rank harmonic amplitude in Fourierseries119888119904 119888119903 and 119888119892 are mesh dampings which can be expressedas follows

119888119898 = 2120589radic 119896119898119898119898119898119899119898119898 + 119898119899 119898 or 119899 = 119904 119903 119901 119892 (8)

where 120589 is gear mesh damping ratio (120589 = 003ndash017) and 119898119898and119898119899 are masses of two meshing gears

As shown in Figure 7 transmission error 119890119899 is approx-imated as superposition of harmonic function of meshfrequency and rotation frequency of shaft [15]

119890119899 = 05119865119901 sin (2120587120596119904119905 + 120593119904) + 051198911015840119901 sin (2120587120596119898119905 + 120593119898)119899 = 119904 119903 119892 (9)

6 Shock and Vibration

Table 1 Technical parameters of TBM cutterhead driving system

Driving motor Rated power 160 kWSpeed range 0ndash1480 rpm

Transmission system Reducer Gear ratio 119894I = 512Ring-pinion Gear ratio 119894II = 126

CutterheadRated power 1600 kW (10lowast160 kW)Speed range 0ndash21 rpmndash47 rpmRated torque 7230KNm 21 rpm

Table 2 Parameters of three-phase asynchronous motor

Parameters ValueRated power 119875119873 160 kWRated voltage 119880119873 400VRated frequency 119891119873 50HzStator resistance 119877119904 001379ΩRotor resistance 119877119903 0007728ΩStator inductance 119871 119904 0152mHRotor inductance 119871 119903 0152mHMutual inductance 119871119898 769mHRotational inertia 119869 29 kgsdotm2

where 119865119901 is total cumulative pitch error 1198911015840119901 is tangentialtolerance of single tooth 120596119904 and 120596119898 are rotation frequencyand mesh frequency and 120593119904 and 120593119898 are initial phase of shaftand mesh phase

3 Dynamic Analysis of ElectromechanicalCoupling Model of CDS

31 Actual Driving Torque of DTC System The technicalparameters of one certain CDS are shown in Table 1According to these parameters the model of three-phaseasynchronous motor is chosen as Table 2 shows and thecontrol parameters of DTC system are set In this paper themultiple invertermotors are supposed to be synchronous andTBM cutterhead is chosen to work under the rotational speed119899119888 = 21 rpm Thus load torque of motor can be calculatedbased on the mean value of load torque on cutterhead whichcan be expressed as (10) shows

119879119871 = 9549 119875119873119894I119894II119899119888119899 (10)

where 119875119873 is rated power 119894I is gear ratio of reducer 119894II is gearratio of ring-pinion gears 119899119888 is rated speed of cutterhead and119899 is number of pinions

Field test data of external load torque is shown in Figure 9In actual tunneling process load torque 119879119871 is unstable andchanges abruptly as geological condition varies On the basisof (10) rated 119879119871 is 1120Nsdotm under rated rotational speed119899119888 = 21 rpm which corresponds to the actual 119879119871 near 310 sin Figure 8 Thus taking a 1 s-length (of) actual 119879119871 between3142 s and 3152 s as an example 119879119871 in the first 02 s keepsstable near rated torque and then rises sharply to 1700Nsdotm at3144 s After 3145 s119879119871 remains roughly stable near 1700Nsdotm

010

020

030

040

050

060

070

080

090

010

0011

0012

0013

0014

0015

00

0500

1000150020002500300035004000

3142 3144 3146 3148 31510001200140016001800

Time t (s)

Actu

al lo

ad to

rque

TL

(Nmiddotm

)

Figure 8 Field test data of external load torque

0 02 04 06 08 1 12 14 16 18 20

500

1000

1500

2000

2500

1 15 2

1000

1500

2000

DTC driving torqueActual driving torque

Time t (s)

Elec

trom

agne

t tor

queT

e(Nmiddotm

)

Figure 9 Actual driving torque of DTC system

with little fluctuations To study the operating charactersof inverter motor under shocking load the 1 s-length of 119879119871between 3142 s and 3152 s is chosen to be simulated as apiecewise function In DTC driving system load torque 119879119871is simulated for 2 s 119879119871 is set to be 1100Nsdotm before 135 s and119879119871 is equal to 1700Nsdotm during 135 s and 2 s

The actual driving torque of DTC system is obtained andshown in Figure 9 In the start-up phase inverter motoroperates with the maximum torque to accelerate to therated speed quickly After operating for 1 s electromagnetictorque 119879119890 fits the actual load torque under rated speedThe fitting result shows that DTC driving system respondsquickly according to the changing load torque 119879119871 Howeverelectromagnetic torque 119879119890 has high torque ripple which isabout 120Nsdotm which can be expressed in discrete form asfollows [16]

119879(119896+1)119890 = 119879(119896)119890 + Δ119879(119896)1198901 + Δ119879(119896)1198901Δ119879(119896)1198901 = minus119879(119896)119890 (119877119904119871 119904 +

119877119903119871119903)119879119904120590

Δ119879(119896)1198902 = 32119899119901 119871119898120590119871 119904119871119903 [(119906(119896)119904 minus 119895120596(119896)119903 120595(119896)119904 ) sdot 119895120595(119896)119903 ] 119879119904(11)

Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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Shock and Vibration 5

119868(3)1199013 (3)1199013 = 119896(3)1199043 119909(3)1199043 119903(3)1199013 minus 119896(3)1199033 119909(3)1199033 119903(3)1199013 + 119888(3)1199043 (3)1199043 119903(3)1199013minus 119888(3)1199033 (3)1199033 119903(3)1199013

119868(3)1199014 (3)1199014 = 119896(3)1199044 119909(3)1199044 119903(3)1199014 minus 119896(3)1199034 119909(3)1199034 119903(3)1199014 + 119888(3)1199044 (3)1199044 119903(3)1199014minus 119888(3)1199034 (3)1199034 119903(3)1199014

119868(3)119888 (3)119888 = 4sum119895=1

[(119896(3)119904119895 119909(3)119904119895 + 119896(3)119903119895 119909(3)119903119895 ) 119903(3)119888 cos120572]

+ 4sum119895=1

[(119888(3)119904119895 (3)119904119895 + 119888(3)119903119895 (3)119903119895 ) 119903(3)119888 cos120572] minus 119896(3)119888 120579(3)119888minus 119888(3)119888 (3)119888 minus 1198963 (120579(3)119888 minus 1205791198921) minus 1198883 ((3)119888 minus 1198921)

11986811989211198921 = 1198963 (120579(3)119888 minus 1205791198921) + 1198883 ((3)119888 minus 1198921) minus 119896119892 (11990311989211205791198921minus 11990311989221205791198922 + 119890119892) minus 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)

11986811989221198922 = 119899 [119896119892 (11990311989211205791198921 minus 11990311989221205791198922 + 119890119892)+ 119888119892 (11990311989211198921 minus 11990311989221198922 + 119890119892)] minus 119879out

(5)

where 119868119904 119868119901 119868119888 1198681198921 and 1198681198922 are mass moments of inertia ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119903119904 119903119901 119903119888 1199031198921 and 1199031198922 are base radiuses of sungear planet gear planet carrier in reducer and pinion-ringgears 120579119904 120579119901 120579119888 1205791198921 and 1205791198922 are angular displacements ofsun gear planet gear planet carrier in reducer and pinion-ring gears 119879in is driving torque of inverter motor which isequal to electromagnet torque 119879119890 in DTC system 119879out is theenlarged driving torque by gear transmission system 119896119888 istorsional stiffness of planet carrier 1198961 1198962 and 1198963 are torsionalstiffnesses of each stage connecting stage 119888119888 is torsionaldamping of planet carrier 1198881 1198882 and 1198883 are torsional dampingsof each stage connecting stage 120572 is pressure angle at the pitchcylinder 119899 is number of pinions 119909119904 is displacement along themeshing line between the sun gear and each planet gear and119909119903 is displacement along the meshing line between the ringgear and each planet gear119909119904 and 119909119903 can be expressed as follows

119909(119894)119904119895 = 119903(119894)119904 120579(119894)119904 minus 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119904119895119909(119894)119903119895 = 119903(119894)119901119895120579(119894)119901119895 minus 119903(119894)119888 120579(119894)119888 cos120572 + 119890(119894)119903119895

(119894 = 1 2 3 119895 = 1 2 3 4) (6)

where 119890119904 is transmission error between the sun gear and eachplanet gear and 119890119903 is transmission error between the ring gearand each planet gear

0 001 002 003 004 005 006 007 008 00913

14

15

16

17

18

19

2

21

Mes

h sti

ffnes

s (N

m)

Time (s)

times109

120596m = 167

km = 175 times 109

Figure 6 Time-varying mesh stiffness

0 05 1 15 2 25 3minus2

minus15

minus1

minus05

0

05

1

15

2

Time (s)

Mes

h er

ror (

m)

times10minus5

120596m = 647

120596s = 94

Fp = 317 times 10minus5

f998400p = 171 times 10minus5

Figure 7 Transmission mesh error

As shown in Figure 6 119896119904 119896119903 and 119896119892 are time-variantmeshstiffnesses which can be expressed by means of the Fourierseries expansion as follows [14]

119896119898 (119905) = 119896119898 + 119873sum119899=1

119861119899 cos 119894120596119898 (119905 + 120593) 119898 = 119904 119903 119892 (7)

where 119896119898 is average mesh stiffness which can be obtainedbased on gear standards such as AGMA ISO 1328-1 andDIN3990 and119861119899 is the n-rank harmonic amplitude in Fourierseries119888119904 119888119903 and 119888119892 are mesh dampings which can be expressedas follows

119888119898 = 2120589radic 119896119898119898119898119898119899119898119898 + 119898119899 119898 or 119899 = 119904 119903 119901 119892 (8)

where 120589 is gear mesh damping ratio (120589 = 003ndash017) and 119898119898and119898119899 are masses of two meshing gears

As shown in Figure 7 transmission error 119890119899 is approx-imated as superposition of harmonic function of meshfrequency and rotation frequency of shaft [15]

119890119899 = 05119865119901 sin (2120587120596119904119905 + 120593119904) + 051198911015840119901 sin (2120587120596119898119905 + 120593119898)119899 = 119904 119903 119892 (9)

6 Shock and Vibration

Table 1 Technical parameters of TBM cutterhead driving system

Driving motor Rated power 160 kWSpeed range 0ndash1480 rpm

Transmission system Reducer Gear ratio 119894I = 512Ring-pinion Gear ratio 119894II = 126

CutterheadRated power 1600 kW (10lowast160 kW)Speed range 0ndash21 rpmndash47 rpmRated torque 7230KNm 21 rpm

Table 2 Parameters of three-phase asynchronous motor

Parameters ValueRated power 119875119873 160 kWRated voltage 119880119873 400VRated frequency 119891119873 50HzStator resistance 119877119904 001379ΩRotor resistance 119877119903 0007728ΩStator inductance 119871 119904 0152mHRotor inductance 119871 119903 0152mHMutual inductance 119871119898 769mHRotational inertia 119869 29 kgsdotm2

where 119865119901 is total cumulative pitch error 1198911015840119901 is tangentialtolerance of single tooth 120596119904 and 120596119898 are rotation frequencyand mesh frequency and 120593119904 and 120593119898 are initial phase of shaftand mesh phase

3 Dynamic Analysis of ElectromechanicalCoupling Model of CDS

31 Actual Driving Torque of DTC System The technicalparameters of one certain CDS are shown in Table 1According to these parameters the model of three-phaseasynchronous motor is chosen as Table 2 shows and thecontrol parameters of DTC system are set In this paper themultiple invertermotors are supposed to be synchronous andTBM cutterhead is chosen to work under the rotational speed119899119888 = 21 rpm Thus load torque of motor can be calculatedbased on the mean value of load torque on cutterhead whichcan be expressed as (10) shows

119879119871 = 9549 119875119873119894I119894II119899119888119899 (10)

where 119875119873 is rated power 119894I is gear ratio of reducer 119894II is gearratio of ring-pinion gears 119899119888 is rated speed of cutterhead and119899 is number of pinions

Field test data of external load torque is shown in Figure 9In actual tunneling process load torque 119879119871 is unstable andchanges abruptly as geological condition varies On the basisof (10) rated 119879119871 is 1120Nsdotm under rated rotational speed119899119888 = 21 rpm which corresponds to the actual 119879119871 near 310 sin Figure 8 Thus taking a 1 s-length (of) actual 119879119871 between3142 s and 3152 s as an example 119879119871 in the first 02 s keepsstable near rated torque and then rises sharply to 1700Nsdotm at3144 s After 3145 s119879119871 remains roughly stable near 1700Nsdotm

010

020

030

040

050

060

070

080

090

010

0011

0012

0013

0014

0015

00

0500

1000150020002500300035004000

3142 3144 3146 3148 31510001200140016001800

Time t (s)

Actu

al lo

ad to

rque

TL

(Nmiddotm

)

Figure 8 Field test data of external load torque

0 02 04 06 08 1 12 14 16 18 20

500

1000

1500

2000

2500

1 15 2

1000

1500

2000

DTC driving torqueActual driving torque

Time t (s)

Elec

trom

agne

t tor

queT

e(Nmiddotm

)

Figure 9 Actual driving torque of DTC system

with little fluctuations To study the operating charactersof inverter motor under shocking load the 1 s-length of 119879119871between 3142 s and 3152 s is chosen to be simulated as apiecewise function In DTC driving system load torque 119879119871is simulated for 2 s 119879119871 is set to be 1100Nsdotm before 135 s and119879119871 is equal to 1700Nsdotm during 135 s and 2 s

The actual driving torque of DTC system is obtained andshown in Figure 9 In the start-up phase inverter motoroperates with the maximum torque to accelerate to therated speed quickly After operating for 1 s electromagnetictorque 119879119890 fits the actual load torque under rated speedThe fitting result shows that DTC driving system respondsquickly according to the changing load torque 119879119871 Howeverelectromagnetic torque 119879119890 has high torque ripple which isabout 120Nsdotm which can be expressed in discrete form asfollows [16]

119879(119896+1)119890 = 119879(119896)119890 + Δ119879(119896)1198901 + Δ119879(119896)1198901Δ119879(119896)1198901 = minus119879(119896)119890 (119877119904119871 119904 +

119877119903119871119903)119879119904120590

Δ119879(119896)1198902 = 32119899119901 119871119898120590119871 119904119871119903 [(119906(119896)119904 minus 119895120596(119896)119903 120595(119896)119904 ) sdot 119895120595(119896)119903 ] 119879119904(11)

Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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6 Shock and Vibration

Table 1 Technical parameters of TBM cutterhead driving system

Driving motor Rated power 160 kWSpeed range 0ndash1480 rpm

Transmission system Reducer Gear ratio 119894I = 512Ring-pinion Gear ratio 119894II = 126

CutterheadRated power 1600 kW (10lowast160 kW)Speed range 0ndash21 rpmndash47 rpmRated torque 7230KNm 21 rpm

Table 2 Parameters of three-phase asynchronous motor

Parameters ValueRated power 119875119873 160 kWRated voltage 119880119873 400VRated frequency 119891119873 50HzStator resistance 119877119904 001379ΩRotor resistance 119877119903 0007728ΩStator inductance 119871 119904 0152mHRotor inductance 119871 119903 0152mHMutual inductance 119871119898 769mHRotational inertia 119869 29 kgsdotm2

where 119865119901 is total cumulative pitch error 1198911015840119901 is tangentialtolerance of single tooth 120596119904 and 120596119898 are rotation frequencyand mesh frequency and 120593119904 and 120593119898 are initial phase of shaftand mesh phase

3 Dynamic Analysis of ElectromechanicalCoupling Model of CDS

31 Actual Driving Torque of DTC System The technicalparameters of one certain CDS are shown in Table 1According to these parameters the model of three-phaseasynchronous motor is chosen as Table 2 shows and thecontrol parameters of DTC system are set In this paper themultiple invertermotors are supposed to be synchronous andTBM cutterhead is chosen to work under the rotational speed119899119888 = 21 rpm Thus load torque of motor can be calculatedbased on the mean value of load torque on cutterhead whichcan be expressed as (10) shows

119879119871 = 9549 119875119873119894I119894II119899119888119899 (10)

where 119875119873 is rated power 119894I is gear ratio of reducer 119894II is gearratio of ring-pinion gears 119899119888 is rated speed of cutterhead and119899 is number of pinions

Field test data of external load torque is shown in Figure 9In actual tunneling process load torque 119879119871 is unstable andchanges abruptly as geological condition varies On the basisof (10) rated 119879119871 is 1120Nsdotm under rated rotational speed119899119888 = 21 rpm which corresponds to the actual 119879119871 near 310 sin Figure 8 Thus taking a 1 s-length (of) actual 119879119871 between3142 s and 3152 s as an example 119879119871 in the first 02 s keepsstable near rated torque and then rises sharply to 1700Nsdotm at3144 s After 3145 s119879119871 remains roughly stable near 1700Nsdotm

010

020

030

040

050

060

070

080

090

010

0011

0012

0013

0014

0015

00

0500

1000150020002500300035004000

3142 3144 3146 3148 31510001200140016001800

Time t (s)

Actu

al lo

ad to

rque

TL

(Nmiddotm

)

Figure 8 Field test data of external load torque

0 02 04 06 08 1 12 14 16 18 20

500

1000

1500

2000

2500

1 15 2

1000

1500

2000

DTC driving torqueActual driving torque

Time t (s)

Elec

trom

agne

t tor

queT

e(Nmiddotm

)

Figure 9 Actual driving torque of DTC system

with little fluctuations To study the operating charactersof inverter motor under shocking load the 1 s-length of 119879119871between 3142 s and 3152 s is chosen to be simulated as apiecewise function In DTC driving system load torque 119879119871is simulated for 2 s 119879119871 is set to be 1100Nsdotm before 135 s and119879119871 is equal to 1700Nsdotm during 135 s and 2 s

The actual driving torque of DTC system is obtained andshown in Figure 9 In the start-up phase inverter motoroperates with the maximum torque to accelerate to therated speed quickly After operating for 1 s electromagnetictorque 119879119890 fits the actual load torque under rated speedThe fitting result shows that DTC driving system respondsquickly according to the changing load torque 119879119871 Howeverelectromagnetic torque 119879119890 has high torque ripple which isabout 120Nsdotm which can be expressed in discrete form asfollows [16]

119879(119896+1)119890 = 119879(119896)119890 + Δ119879(119896)1198901 + Δ119879(119896)1198901Δ119879(119896)1198901 = minus119879(119896)119890 (119877119904119871 119904 +

119877119903119871119903)119879119904120590

Δ119879(119896)1198902 = 32119899119901 119871119898120590119871 119904119871119903 [(119906(119896)119904 minus 119895120596(119896)119903 120595(119896)119904 ) sdot 119895120595(119896)119903 ] 119879119904(11)

Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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Shock and Vibration 7

Table 3 Parameters of 3-stage planetary reducer in TBM

Parameter Sun Planet Ring Carrier1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd

Massm (kg) 369 955 1406 237 578 1406 1776 3006 4298 1832 321 4735I (kgsdotm2) 0004 0019 0332 0002 0007 0032 0132 039 088 0145 056 186Tooth number z 25 27 24 20 21 24 65 69 72 mdash mdash mdashModule119898119899 1198981198991 = 4 1198981198992 = 5 1198981198993 = 6Tooth width b (m) 1198871 = 006 1198872 = 0085 1198873 = 011Pressure angle 120572 120572(1)119904 = 120572(2)119904 = 120572(3)119904 = 20∘ 120572(1)119903 = 120572(2)119903 = 120572(3)119903 = 20∘Mesh stiffness 119896119898 (Nm)

119896(1)119904 = 9872 times 108119896(1)119903 = 1179 times 109

119896(2)119904 = 13225 times 109119896(2)119903 = 14366 times 109

119896(3)119904 = 17362 times 109119896(3)119903 = 17543 times 109

Table 4 Natural frequencies of planetary reducer

Motion modes Natural frequency (Hz)Rigid motion mode 1198911 = 0Rotational vibration modes 1198912 = 308 1198913 = 529 1198914 = 2806 1198918 = 3772 1198919 = 4644 11989113 = 5919 11989116 = 6798 11989117 = 8338Planet vibration modes 1198915 = 1198916 = 1198917 = 3598 11989110 = 11989111 = 11989112 = 4965 11989114 = 11989115 = 6655

where 119879(119896+1)119890 and 119879(119896)119890 are electromagnetic torques at 119896 + 1and 119896 moment Δ119879(119896)1198901 is torque attenuation caused by statorand rotor resistance Δ119879(119896)1198902 is torque variation caused byvoltage space vector 119879119904 is sampling time 120590 is constant whichis related to 119871119898 119871 119904 and 119871119903 and 120596119903 is speed of rotor

Based on (11) torque ripple is inevitable and influencedby sampling time motor speed flux and voltage vectorwhich are closely related to computing power of digitalcontroller and switching frequency [17] Therefore as theexternal excitation of gear transmission system torque rippleof electromagnetic torque 119879119890 may be higher in actual motordriving process and influence the dynamic characteristics ofgear transmission system

32 Modal Property of Multistage Gear Transmission SystemIn multistage gear transmission system one-stage pinion-ring gears consist of several pinions 1198921 and one ring gear 1198922The size of ring gear 1198922 is much bigger than other gears andthe inherent properties of planetary reducer cannot be clearlypresented under the influence of ring gear 1198922 Thereforethe modal properties of planetary reducer are chosen to beanalyzed in this paper

Based on (5) equivalent mathematic model of planetaryreducer can be expressed in the form of matrix

119872 (119905) + 119862 (119905) + 119870119902 (119905) = 119865 (119905) (12)

where 119902(119905) is vibration displacement vector119872 ismassmatrix119862 is damping matrix 119870 is stiffness matrix and 119865(119905) isexcitation vector

Since the variation range ofmesh stiffness is not bigmeshstiffness is simplified as average stiffness In the same stage allexternal mesh stiffness and all internal mesh stiffness are thesame separately The influence of damping is also ignored to

obtain the natural frequencies Thus the eigenvalue problemof (12) can be expressed as follows

1205962119894119872120593119894 = 119870120593119894 (13)

where 120596119894 is i-order natural frequency 119870 is average stiffnessmatrix and 120593119894 is i-order vibration mode vector as

120593119894 = [120601(1)119894119904 120601(1)1198941199011 120601(1)1198941199012 120601(1)1198941199013 120601(1)119894119888 120601(2)119894119904 120601(2)1198941199011 120601(2)1198941199012 120601(2)1198941199013 120601(2)1198941199014120601(2)119894119888 120601(3)119894119904 120601(3)1198941199011 120601(3)1198941199012 120601(3)1198941199013 120601(3)1198941199014 120601(3)119894119888 ]

(14)

According to the main parameters of planetary reducerlisted in Table 3 natural frequencies and vibrationmodes canbe obtained by solving (13) Natural frequencies are listed inTable 4 and vibration modes are shown in Figure 10 Basedon the inherent properties planetary reducer operates inthree types of vibrationmodes rigidmotionmode rotationalvibration mode and planet vibration mode In rigid motionmode natural frequency 1198911 = 0Hz and all componentsjust operate on the basis of transmission ratio withoutvibration In rotational vibration mode natural frequenciesf are distinct and f = 0Hz All components have rotationalvibration and planet gears in each stage operate with thesame vibration In planet vibrationmode natural frequencies1198915 = 1198916 = 1198917 = 3805Hz 11989110 = 11989111 = 11989112 = 5266Hzand 11989114 = 11989115 = 7056Hz All central components such assun gears and planet carriers have no vibration except planetgears

33 Dynamic Results of Electromechanical Model

331 Vibration Displacement Vibration displacement is oneof the most important elements in dynamic response whichdenotes the vibration degree of gear transmission system

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

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Shock and Vibration

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DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 5 10 15

051015minus1

minus05

0

05

1

Rela

tive a

mpl

itude

Degree of freedom Natural frequency

Figure 10 Vibration modes of planetary reducer

Based on the parameters listed in Tables 1 2 and 3 vibrationdisplacement can be obtained by solving the electromechan-ical coupling model As shown and discussed above torqueripple of inverter motor is unavoidable and may influencethe dynamic response of gear transmission systemThereforevibration displacements under electromagnetic torque 119879119890with ripple and idealized piecewise torque without ripple arecalculated separately

To ensure the accuracy of results and spare calculationtime Runge-Kutta integration method is chosen to solve theequivalent mathematic model in 1 s Dynamic responses ofsun gears are taken as an example Vibration displacements ofsun gear in each stage are shown in Figure 11 Sun gears vibratenear the equilibrium position and vibration amplitudesdecrease as driving torque rises Vibration amplitude of 2nd-stage sun gear is the smallest and significantly smaller thanthe amplitudes of other sun gears which are approximatelyequal Therefore in the antivibration design process of 3-stage gear transmission system in CDS 1st-stage and 3rd-stage gears should be the primary design targets

For a comparison of dynamic responses under two kindsof driving torque 120579119904119890 herein is defined as the vibrationdisplacement of sun gear under electromagnetic torque 119879119890and 120579119904119898 herein is defined as the vibration displacement ofsun gear under idealized piecewise torque In the case of 1st-stage sun gear for 035 s and 1 s mean values of 120579119904119890 and 120579119904119898are the same and equal to 00286 which means that actualdriving torque of inverter motor has no effect on equilibriumposition However standard deviation of 120579119904119890 is 00092 andstandard deviation of 120579119904119898 is 00045 which indicates thatthe vibration amplitude under electromagnetic torque 119879119890 isbigger than the one under idealized piecewise torque Thusit is tempting to conclude that the actual driving torque ofinverter motor may aggravate vibration of gear transmissionsystem owing to the torque ripple

332 Dynamic Meshing Force Dynamic meshing force di-rectly influences the failure of gear transmission system

such as wear or pitting of gear teeth Meshing force can beexpressed based on (1) as follows

119865(119894)119904119895 = 119896(119894)119904119895 119909(119894)119904119895 + 119896119888(119894)119904119895 (119894)119904119895119865(119894)119903119895 = 119896(119894)119903119895 119909(119894)119903119895 + 119888(119894)119903119895 (119894)119903119895

(15)

where 119865119904 and 119865119903 are externalinternal meshing forces 119896119904 and119896119903 are time-variant mesh stiffnesses 119909119904 is displacement alongthe meshing line between the sun gear and each planet gearand 119909119903 is displacement along the meshing line between thering gear and each planet gear

Under the external excitation of electromagnetic torque119879119890 dynamic meshing forces in each stage are calculated and apart of them are shown in Figures 13 and 14 In time domainexternal meshing forces increase abruptly as electromagnetictorque 119879119890 changes at 035 s and meshing forces increase bystage according to gear ratio Meshing forces of 1st-stageplanet gears fluctuate more apparently than the other twostages at changing point which can be probably attributed tothe fact that 1st-stage sun gear is directly under the influenceof external excitation In the same stage meshing forces ofplanet gears are also different from each other As shown inFigure 12 load-sharing level of 3rd stage is the highest andload-sharing level of 1st stage is the lowest which may becaused by phase difference ofmesh stiffness and transmissionerror

Spectral analysis of externalmeshing force in each stage isshown in Figure 13 Herein 119891119899119894 (119894 = 2 3) donates the i-ordernatural frequency and 119891119898119895 (119895 = 1 2 3) donates the j-stagemesh frequency As shown in Figure 13 meshing forces ineach stage vibrate in the low frequency domain which is near119891119898119895 and its multiple frequencies Furthermore low-ordernatural frequency (1198911198992 = 308 1198911198993 = 529) also exist in theinternal excitations and 1198911198992 possesses the largest amplitude

4 Further Discussion

As shown in Figure 11 vibration of gear transmission systemis increased under electromagnetic torque 119879119890 compared withidealized driving torque The increases of vibration on eachcomponent may be related to electromagnetic torque 119879119890 andits torque ripple To assess the impact of electromagnetictorque 119879119890 on each componentrsquos vibration an influence index120575 of torque ripple is proposed based on the vibration displace-ments as (16) expresses

120575 = 119860119890 minus 119860119898119860119898max (16)

where 119860119890 and 119860119898 denote the deviation value from equi-librium position under electromagnetic torque 119879119890 and ide-alized torque respectively 119860119898max is the maximum of 119860119898which represents vibration degree and 119860119890119894 and 119860119898119894 can beexpressed as follows

119860 119904 = 10038161003816100381610038161003816 120579119904 minus 12057911990410038161003816100381610038161003816 (119904 = 119890119898) (17)

where 120579119904 is the vibration displacement of one componentunder electromagnetic torque 119879119890 and idealized torque and 120579119904is mean value of 120579119904 which represents equilibrium position

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

0 02 04 06 08 1minus001

0

001

002

003

004

005

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(a) 1st-stage sun gear

0 02 04 06 08 1minus6

minus5

minus4

minus3

minus2

minus1

0

1

2

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

times10minus3

Time t (s)

(b) 2nd-stage sun gear

0 02 04 06 08 1minus005

minus004

minus003

minus002

minus001

0

001

Disp

lace

men

t (Ra

d)

Under actual torque with rippleUnder idealized torque without ripple

Time t (s)

(c) 3rd-stage sun gear

Figure 11 Dynamic response of sun gears

Influence index 120575 of torque ripple on all the componentsis calculated under actual driving torque shown in Figure 9120575 on different components in the same stage are shown andcompared in Figure 14 120575 on the same component in differentstages are shown and compared in Figure 15 In time domain120575 on all the components are greater than zero which meansvibrations of all the components are aggravated by torqueripple of electromagnetic torque 119879119890 To each component120575 increases as load torque 119879119871 changes from 1100Nsdotm to1700Nsdotm In the same stage 120575 on sun gear is the largest andthe impact of electromagnetic torque 119879119890 on planet carrier isthe smallest In different stages 120575 on sun gear in 2nd stage isthe smallest and the impacts on sun gears in 1st stage and 3rdstage are similarThus as an important performancemeasurethe influence index 120575 on sun gear in 1st stage or 3rd stage canbe taken as the optimization objective tominimize the impactof torque ripple

To study the impact of torque ripple on vibration furthera series of electromagnetic torque 119879119890 with different torque

ripples are simulated as load torque 119879119871 is 1700Nsdotm anddynamic responses under such torques are obtained Maxi-mumof influence index120575max is chosen to represent the overallimpact of electromagnetic torque 119879119890 with different torqueripples and 120575max on all components are shown in Figure 16It can be seen that vibration degrees of all the componentsare aggravated more severely as torque ripple increases andtendencies of the impact on each component are similarTherefore the ripple of electromagnetic torque 119879119890 should becontrolled to be as small as possible As shown and discussedabove torque ripple is influenced by several parametersSince the asynchronous motor is chosen according to thetunneling conditions parameters of motor are fixed andcannot be adjusted Thus in the process of optimizingcontrol method of inverter motor torque ripple should bereduced by rectifying parameters of speed controller in DTCsystem Furthermore on the premise of meeting tunnelingrequirements motor speed can be reasonably controlled tominimize the torque ripple

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

0 02 04 06 08 1minus05

0

05

1

15

2

25

Planet gear 1Planet gear 2

Planet gear 3

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(a) 1st stage

0 02 04 06 08 1minus1

0

1

2

3

4

5

6

7

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(b) 2nd stage

0 02 04 06 08 1minus2

02468

10121416

Planet gear 1Planet gear 2

Planet gear 3Planet gear 4

times104

Time t (s)

Exte

rnal

mes

hing

forc

eFs

(N)

(c) 3rd stage

Figure 12 External meshing force in time domain

5 Conclusion

In this paper an electromechanical coupling model ofTBM cutterhead driving system (CDS) is established whichincludes a simulation model of direct torque control (DTC)driving system and a purely torsional dynamic model ofmultistage gear transmission system Based on this couplingmodel electromagnetic torque 119879119890 is obtained and torque rip-ple is analyzedModal property and dynamic response of geartransmission system are calculated and the impact of torqueripple on vibration is analyzed The specific conclusions ofthis study are as follows

(1) DTC driving system responds quickly as load torquechanges and electromagnetic torque119879119890 has high torque ripplewhich is about 120Nsdotm Torque ripple is influenced bysampling time motor speed flux and voltage vector whichare closely related to computing power of digital controllerand switching frequency

(2) Based on the dynamic analysis of gear transmissionsystem vibration modes of transmission system can beclassified into three types rigid motion mode rotationalvibrationmode and planet vibrationmode For a comparisonof vibration displacements the vibration amplitude of 2nd-stage component is the smallest among all the three stagesMeshing forces mainly vibrate in the low frequency domainwhich approaches to mesh frequency and low-order naturalfrequency (1198912 = 308Hz 1198913 = 529Hz) Moreover meshingforces increase by stage according to gear ratio and meshingforces of 1st-stage planet gears fluctuate more apparently thanthe other two stages at changing point of load torque

(3) Compared with the dynamic responses under ide-alized piecewise torque vibration displacements of geartransmission system under electromagnetic torque 119879119890 areaggravated owing to the torque ripple Dynamic index 120575is proposed and discussed to show the impact of electro-magnetic torque 119879119890 In the same stage 120575 of sun gear is

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0 1 2 3 4 50

2000

4000

6000

8000

10000

12000

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5001000150020002500

times104

fn2

fm1fn3

(a) 1st stage

0 1 2 3 4 50

05

1

15

2

25

3

35

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

5000

10000

times104

times104

fm2 fn2fn3

2fm2 3fm2

(b) 2nd stage

0 1 2 3 4 50

1

2

3

4

5

6

7

8

Frequency f (Hz)

Am

plitu

de (N

)

0 500 1000 1500 20000

1

2

3

times104

times104

times104

fm3

fn2

fn3

2fm3

3fm3

(c) 3rd stage

Figure 13 External meshing force in frequency domain

0 02 04 06 08 10

05

1

15

Sun gearPlanet gearPlanet carrier

Time t (s)

Influ

ence

inde

x120575

Figure 14 Influence index 120575 on different components in the 1ststage

the largest The impact on 2nd-stage components is thesmallest in different stages Furthermore vibration degreesof all the components are aggravated more severely as torque

0 02 04 06 08 10

05

1

15

1st stage3rd stage2nd stage

Time t (s)

Influ

ence

inde

x120575

Figure 15 Influence index 120575 on sun gear in different stages

ripple increases Thus torque ripple should be minimized byoptimizing the control method of inverter motor

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

Stage

1 152

253 50

100150

200

0

1

2

3

Sun gearPlanet gearPlanet carrier

Torque ripple ΔTe (Nmiddotm)

Influ

ence

inde

x120575

max

minus1

Figure 16 Influence index 120575max under different torque ripples

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

TheNationalNatural Science Foundation of China (Grant no51605071) National Basic Research Program (973 Program)of China (Grant no 2013CB035400) China PostdoctoralScience Foundation (Grant no 2015M570245) and OpenFoundation of the State Key Laboratory of Fluid PowerTransmission and Control of Zhejiang University of China(Grant no GZKF- 201414) are acknowledged for their finan-cial supports

References

[1] W Sun J Ling J Huo L Guo X Zhang and L DengldquoDynamic characteristics study with multidegree-of-freedomcoupling in TBM cutterhead system based on complex factorsrdquoMathematical Problems in Engineering vol 2013 Article ID635809 17 pages 2013

[2] A Delisio J Zhao and H H Einstein ldquoAnalysis and predic-tion of TBM performance in blocky rock conditions at theLotschberg Base Tunnelrdquo Tunnelling and Underground SpaceTechnology vol 33 pp 131ndash142 2013

[3] A Kahraman ldquoLoad sharing characteristics of planetary trans-missionsrdquo Mechanism and Machine Theory vol 29 no 8 pp1151ndash1165 1994

[4] J Huo H Wu G Li W Sun and J Chen ldquoThe couplingdynamic analysis and field test of TBM main system undermultipoint impact excitationrdquo Shock and Vibration vol 2015Article ID 313259 14 pages 2015

[5] JWeiQ SunW Sun J Cai and J Zeng ldquoDynamic analysis andload-sharing characteristic of multiple pinion drives in tunnelboring machinerdquo Journal of Mechanical Science and Technologyvol 27 no 5 pp 1385ndash1392 2013

[6] J Wei Q Sun W Sun X Ding W Tu and Q Wang ldquoLoad-sharing characteristic of multiple pinions driving in tunnelingboring machinerdquo Chinese Journal of Mechanical Engineeringvol 26 no 3 pp 532ndash540 2013

[7] W Sun X Ding J Wei X Wang and A Zhang ldquoHierarchicalmodeling method and dynamic characteristics of cutter head

driving system in tunneling boring machinerdquo Tunnelling andUnderground Space Technology vol 52 pp 99ndash110 2016

[8] K Zhang H Yu Z Liu and X Lai ldquoDynamic characteristicanalysis of TBM tunnelling in mixed-face conditionsrdquo Simula-tion Modelling Practice and Theory vol 18 no 7 pp 1019ndash10312010

[9] D Qin and Y Zhao ldquoMulti-objective optimization of multi-stage planetary gear train used in shield machine cutter driverrdquoChina Mechanical Engineering vol 23 no 1 pp 12ndash17 2012

[10] R Liu D Y Yu W G Zhao W D Li and J Z Sun ldquoResearchon adaptive load sharing control for multi-motor synchronousdriving system of shield machinerdquo Applied Mechanics andMaterials vol 667 pp 417ndash420 2014

[11] R Liu J Z Sun Y Q Luo W Sun and W D Li ldquoResearchonmulti-motor synchronization control based on the ring cou-pling strategy for cutterhead driving systemof shieldmachinesrdquoApplied Mechanics and Materials vol 52ndash54 pp 65ndash72 2011

[12] J Z Sun R Liu Y Q Luo and W Sun ldquoResearch onmulti-motor synchronization control for cutter head of shieldmachine based on the ring coupled control strategyrdquo in Intel-ligent Robotics and Applications vol 5928 of Lecture Notes inComputer Science pp 345ndash354 Springer 2009

[13] H Y Kanaan K Al-Haddad and G Roy ldquoAnalysis of theelectromechanical vibrations in induction motor drives dueto the imperfections of the mechanical transmission systemrdquoMathematics and Computers in Simulation vol 63 no 3ndash5 pp421ndash433 2003

[14] RG Parker and J Lin ldquoMesh phasing relationships in planetaryand epicyclic gearsrdquo Journal of Mechanical Design vol 126 no2 pp 365ndash370 2004

[15] DQin Z Xiao and JWang ldquoDynamic characteristics ofmulti-stage planetary gears of shield tunneling machine based onplanet mesh phasing analysisrdquo Journal of Mechanical Engineer-ing vol 47 no 23 pp 20ndash29 2011

[16] D Casadei G Serra and A Tani ldquoImplementation of adirect torque control algorithm for induction motors based ondiscrete space vector modulationrdquo IEEE Transactions on PowerElectronics vol 15 no 4 pp 769ndash777 2000

[17] T Noguchi M Yamamoto S Kondo and I Takahashi ldquoHighfrequency switching operation of PWM inverter for directtorque control of induction motorrdquo in Proceedings of the IEEEIndustry Applications Conference 32nd IASAnnualMeeting Part3 (of 3) pp 775ndash780 October 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of