Research ArticleDynamic Analysis of Integrally Geared Compressors withVarying Workloads

Ming Zhang Zhinong Jiang and Jinji Gao

Diagnosis and Self-Recovery Engineering Research Center Beijing University of Chemical Technology Beijing 100029 China

Correspondence should be addressed to Zhinong Jiang jiangzhinong263net

Received 19 June 2016 Revised 22 August 2016 Accepted 20 September 2016

Academic Editor Chao Tao

Copyright copy 2016 Ming Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Integrally geared compressors are characterized by compact and high efficiency machines which are widely used in modernprocessing industries As an important part of integrally geared compressors a geared rotor-bearing system exhibits complicateddynamic behaviors When running at rated speeds a coupling system likely produces resonance with an adjusted workload and acritical load phenomenon occursThe dynamic coefficients of bearings axial force and torque and gear meshing stiffness vary withworkload because of the interaction between rotors In this study a dynamic model of a geared rotor-bearing system influencedby the dynamic coefficients of bearings axial force and torque and gear meshing stiffness is developed The dynamic responsesof the coupling system are calculated and analyzed by using a typical five-shaft integrally geared compressor as an example Theeffects of different parameters on the dynamic behaviors of the proposed system are also considered in the discussion The gearedrotor-bearing system is further investigated to examine the failure mechanism of the critical load

1 Introduction

Integrally geared compressors are key components of pro-cessing industries They are applied extensively in refinerieschemical plants and other fields because of their compactstructure high efficiency and wide performance Gearedrotor-bearing systems are generally operated beyond the firstor second critical speed and such systems provide secure andsteady operation that contributes to the safety stability andlong-term operation of integrally geared compressors Thesesystems can also be considered a set of rotor-bearing systemsdynamically interacting with one another and exhibitingvery complicated dynamic behaviors [1] Therefore accu-rate dynamic knowledge about geared rotor-bearing systemsshould be obtained

Geared rotor-bearing systems can be efficiently modeledby finite element methods Nelson and McVaugh [2] used aconsistent matrix approach and developed a finite elementmodel of a rotor-bearing system by considering the effectsof rotatory inertia gyroscopic moments and axial forceZorzi and Nelson [3 4] increased the influence of internaldamping and axial torque on the basis of a previous model[2] Kahraman et al [5] creatively utilized the finite element

method to establish a model of a geared rotor-bearing systemby determining the rotary inertia of a shaft element axialforce on shafts flexibility and damping of bearings materialdamping of shafts and stiffness and damping of a gear meshTheir work [2ndash5] illustrated that accurate dynamic behaviorscan be obtained with finite element method

Compared with general rotor-bearing systems gearedrotor-bearing systems are difficult to analyze Neverthelessgeared system dynamics have been extensively investigatedRao et al [6] examined the coupling between bending andtorsion caused by gears and considered the axial torque todetermine the lateral response to torsional excitation Wuand Chen [7] presented a simple approach to eliminatetorsional vibration in a gear-branched system via finiteelement method Choi et al [8] and Lee et al [9] evaluatedthe dynamic characteristics of coupled lateral torsional andaxial vibration in a helical geared system Kubur et al [10]and Lee and Ha [11] also analyzed the unbalance response ofa geared rotor-bearing system A dynamic model of gearedrotor-bearing systems has also been developed on the basisof previous studies and relevant factors including geomet-ric eccentricity transmission error and some unavoidabledefects [12ndash14] The nonlinear dynamic characteristics of

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 2594635 13 pageshttpdxdoiorg10115520162594635

2 Shock and Vibration

geared rotor-bearing systems have also been widely explored[15ndash17]

The complicated dynamic behaviors including linear andnonlinear characteristics of geared rotor-bearing systemshave been extensively investigated However the dynamiccharacteristics of geared rotor-bearing systems consideringvaryingworkload have been rarely explored Integrally gearedcompressors likely cross numerous critical speeds duringcompressor start up When running at rated speeds inte-grally geared compressors produce resonance with adjustedworkload and a critical load phenomenon occurs Gao[18] discovered and successfully resolved a DH integrallygeared compressor failure caused by critical load Thereforeour study mainly aims to examine the effect of differentworkloads on the dynamic characteristics of a geared rotor-bearing system and to investigate the failure mechanism ofthe critical load

In this paper a general finite element approach of gearedrotor-bearing system is presented in Section 2 The dynamicmodel of geared rotor-bearing system covers the effects ofbearing flexibility and damping axial force and torque andgear meshing A five-shaft integrally geared compressor ismodeled and discussed in Section 3 The bearing dynamiccoefficients axial force and torque and gearmeshing stiffnessare changed as workload varies because of the interaction inthe geared rotor-bearing system The dynamic responses ofthe system under different parameters are also analyzed Ourresults indicate that the bearing dynamic coefficients and gearmeshing stiffness significantly influence the responses of thegeared rotor-bearing system but the axial force and torqueslightly influence such responses The resonance peaks andtheir phase of the critical load are obtained from the dynamicresponses affected by the bearing dynamic coefficients axialforce and torque and gear meshing stiffness

2 Dynamic Modeling for GearedRotor-Bearing System

In this section the FE dynamic equation of the geared rotor-bearing system influenced by the flexibility and dampingof bearings axial force axial torque and gear meshing ispresented The assembly method of the coupling systemequation is also investigated

21 Rotor-Bearing System Dynamic Equation The coordi-nates defining bending as the two planes are shown inFigure 1 The local coordinate of the finite rotor element is119902119890 = [119906119894 V119894 120579119894 120595119894 119906119895 V119895 120579119895 120595119895] The dynamic equation of thefinite rotor element is as follows

(119872119890119879 +119872119890119877) 119890 minus Ω119866119890119890 + 119870119890119861119902119890 = 119876119890 (1)

where 119872119890119879 and 119872119890119877 are the element mass matrices 119870119890119861 isthe element bending stiffness matrix 119866119890 is the elementgyroscopicmatrixΩ is the spin speed119876119890 is the external forcevector of the rotor element The matrices in (1) are defined inAppendix A

The coupling of the input shaft the impellers of outputshafts and the gear of the middle shaft are regarded as a rigid

disk (with gyroscopic effect) When the local coordinates aredefined as 119902119889 = [119906 V 120579 120595] the dynamic equation of the rigiddisk is as follows

119872119889119889 minus Ω119866119889119889 = 119876119889 (2)

where Ω is the spin speed 119876119889 is the external force vector ofthe rigid discs119872119889 and119866119889 are themassmatrix and gyroscopicmatrix for the disk respectively and defined as follows

119872119889 =[[[[[[

119898119889 0 0 00 119898119889 0 00 0 119868119889 00 0 0 119868119889

]]]]]]

119866119889 =[[[[[[

0 0 0 00 0 0 00 0 0 1198681199010 0 minus119868119901 0

]]]]]]

(3)

The bearings utilized in this paper are limited to thosewhich obey the governing equations of the form

119862119887119887 + 119870119887119902119887 = 119876119887 (4)

in the local coordinates where119876119887 is the bearing external forcevector 119870119887 = [ 119896119906119906 119896119906V119896V119906 119896VV

] is the stiffness matrix of bearing and119862119887 = [ 119888119906119906 119888119906V119888V119906 119888VV ] is the damping matrix of bearing

22 Incremental Stiffness due to Axial Force and Axial TorqueIn the integrally geared compressor the mechanical powerdeveloped by the turbine is transmitted through the inputshaft to the output shaft impellers where gas is pressurizedThe purpose of the geared rotor-bearing system usuallydepends fundamentally on this transmission of torque Eachshaft of the compressor carries some torque about the axisof rotation The helical gear meshing is generally usedfor integrally geared compressors to ensure the stability oftransmission The mechanical power of integrally gearedcompressors contributes to not only the axial torque but alsothe axial force because of the effect of helical gear meshingThe axial force along the undeformed axis of the rotor occursin each shaft of the coupling system Nelson andMcVaugh [2]developed the equation of motion for a rotating finite shaftelement including the effect of axial force The result is astiffening effect if the shaft is in tension or a softening effect

Shock and Vibration 3

z

ui

120595i

uj

120595j

i

120579i

j

120579j

z

x-z plane

y-z plane

Figure 1 The local coordinates in the bending planes

if the force is compressive The lateral motion of an Euler-Bernoulli beam element of the axial force is as follows

119870119890119860

= 11986511988630119897119890

[[[[[[[[[[[[[[[[[[

360 36 sym

0 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36 3119897119890 0 0 360 minus3119897119890 minus1198972119890 0 0 3119897119890 411989721198903119897 0 0 minus1198972119890 minus3119897119890 0 0 4119897119890

]]]]]]]]]]]]]]]]]]

(5)

where 119865119886 is the axial tensile force within the elementThe axial torque within a shaft may affect the lateral

behavior which is similar to the effect caused by the axialforce in a shaft Relatively few papers or texts on rotordynamics consider this coupling although Zorzi and Nelson[4] derive the contribution to the stiffness for the lateralmotion of an Euler-Bernoulli beam element as follows

119870119890119879 = 1205911198902119897119890

[[[[[[[[[[[[[[[[[[

0 0 2 0 0 0 minus2 00 0 0 2 0 0 0 minus22 0 0 minus119897119890 minus2 0 0 1198971198900 2 119897119890 0 0 minus2 minus119897119890 00 0 minus2 0 0 0 2 00 0 0 minus2 0 0 0 2minus2 0 0 minus119897119890 2 0 0 1198971198900 minus2 119897119890 0 0 2 minus119897119890 0

]]]]]]]]]]]]]]]]]]

(6)

where 120591119890 is the transmitted torque within the element

23 Gear Meshing Stiffness Connecting the shafts throughthe helical gear in the geared rotor-bearing system oftencauses couple vibration Helical gear meshing stiffness isinvestigated on the basis of a previous study [6] The rela-tionship between contact force and displacements of a gearpair is determined by representing the teeth in contact withan equivalent stiffness along the pressure line as shown inFigure 2 The rotations 120579 and 120595 are small in a gear pair assuch we can disregard these terms Along the pressure linegear119894 moves as follows

119909119894119892 = 119906119894 sin120572 cos120573 + V119894 cos120572 cos120573 + 119903119894120601119894 cos120573 (7)

and gear119895 moves as follows

119909119895119892 = 119906119895 sin120572 cos120573 + V119895 cos120572 cos120573 + 119903119895120601119895 cos120573 (8)

where 119903119894 and 119903119895 are the base circle radii of the two gears 120572 isthe pressure angle 120573 is helix angle 120601119894 and 120601119895 are the torsionaldisplacements of the gears about the 119911-axil 119906119894 and 119906119895 are thedisplacement of gear119894 and gear119895 in the 119909 direction V119894 and V119895are the displacement of gear119894 and gear119895 in the 119910 direction

Therefore the relative displacement at the gear mesh is asfollows

119909119892 = (119906119895 minus 119906119894) sin120572 cos120573 + (V119895 minus V119894) cos120572 cos120573minus (119903119894120601119894 + 119903119895120601119895) cos120573 = 119873119892119902119892

(9)

where

119873119892 = [minus sin120572 cos120573 minus cos120572 sin120573 0 0 sin120572 cos120573 cos120572 cos120573 0 0 minus119903119894 cos120573 minus119903119895 cos120573]

119902119892 = [119906119894 V119894 120579119894 120595119894 119906119895 V119895 120579119895 120595119895 120601119894 120601119895] (10)

The strain energy within the gear mesh is as follows

119880119892 = 121198961198921199092119892 = 1

2119896119892119902119879119892119873119879119892119873119892119902119892 (11)

where 119896119892 is the gear mesh stiffness Thus the stiffness matrixof gear mesh is as follows

119870119892 = 119896119892119873119879119892119873119892 (12)

The matrix of119870119892 is defined in Appendix B

4 Shock and Vibration

24 The Dynamic Equation of Geared Rotor-Bearing SystemEquations (1) (2) (5) (6) and (12) can be combined forthe geared coupled system The dynamic equation of gearedrotor-bearing system is as follows

119872119904119904 + (119862119904119887 minus Ω119866119904) 119904+ (119870119904119861 + 119870119904119887 + 119870119904119892 minus 119870119904119860 minus 119870119904119879) 119902119904 = 119876119904 (13)

where 119872119904 is the mass matrix 119862119904119887 is the damping matrix ofbearing Ω is the spin speed 119866119904 is the gyroscopic matrix119870119904119861 119870119904119887 119870119904119892 119870119904119860 and 119870119904119879 are the stiffness matrix of the rotorbearing gear axial load and axial torque respectively 119876119904 isthe external force vector 119902119904 is the displacement vector whichis defined as follows119902119904 = [1199061 V1 1205791 1205951 1206011 1199062 V2 1205792 1205952 1206012 119906119899 V119899 120579119899 120595119899

120601119899] (14)

25 Assembly of the Coupling System Equation In order toconstruct an entire geared rotor-bearing system equationan assembly method of the single rotor finite element (FE)models of the shafts bearings and disks and the gear pairFE models is stated as follows The single rotor FE modelassembly method is detailed in the book by Vollan andKomzsik [19] Figure 3 shows such a simple geared systemwhich is used as an example to illustrate the assemblymethod of the entire system equation The nodes on eachgear are connected by a stiffness matrix In a geared systemthe element matrices are inserted into the system matricesin the positions determined by the positions of the localelement coordinates in the global vectorThis is demonstrateddiagrammatically in Figure 4 for the example of Figure 3 inwhich the nonzero degrees of freedom are shadedThe blocksin shades of light gray and dark gray denote the positionscorresponding to Rotor 1 and Rotor 2 respectively The blacksquares denote the stiffness matrix for the connection of thegear meshing which is split into four blocks that slot into theposition of Node 3 and Node 6 in the entire system equationThen the entire assembled equation of a geared system isimplemented as shown in Figure 4

26 Validation of Dynamic Model A 600 kW turbo-chillerrotor-bearing system with a bull-pinion speed-increasinggear is applied to validate the correctness of the modelproposed in this paper The dynamic model of the turbo-chiller rotor-bearing system is shown in Figure 5 and thesystem parameters are based on a previous study [9] Witha test unbalance of 119880 = 197 gsdotmm attached to the impellerthe coupled unbalance responses at the driver shaft calculatedby the proposed method are shown in Figure 6 Our results(Figure 6) are consistent with those described in a previousstudy [9]

3 Results and Discussions

Applying the proposed method we perform an unbalanceresponse analysis with a five-shaft geared rotor-bearing sys-tem of an integrally geared compressor (Figure 7) The input

ui

i

uj

j

i

i

j

j

i

kgjGear

Gear zi

zj

Ti

Tj

Oi

Oj

i

j

Figure 2 Gear meshing system model

Node 1 Node 2

Node 3

Node 4 Node 5

Node 6

Node 7 Node 8

Rotor 1

Rotor 2

Figure 3 Rotors coupled by a gear

Rotor 1 Rotor 2

Figure 4 Structure of the entire assembled equation for a gearedsystem

Shock and Vibration 5

D1 D2 D3 D4 G1

D5 D6D7 D8B1 B2

B3 B4

Driver shaft

Driven shaft

Figure 5 Dynamic model of the turbo-chiller rotor-bearing systemin [9]

At bull gear (this paper)At bull gear ([9])At motor side bearing (this paper)At motor side bearing ([9])

1500 2000 2500 3000 35001000Rotating speed (rpm)

Resp

onse

(m

pea

k to

pea

k)

0

001

002

003

004

005

006

Figure 6 Unbalance responses comparing with [9]

shaft and three output shafts (O1 O2 and O3) are coupledtogether by the transmission gear of the middle shaft in thiscoupling system Each shaft is a general rotor-bearing systemsupported by oil film bearings The impellers are fixed on theoutput shafts to compress the gas

In the five-shaft geared rotor-bearing system each shafthas the same material properties Youngrsquos modulus of shaftis 119864 = 2059 times 1011Nm2 shear modulus of shaft is 119866 =7919 times 1010Nm2 the density of the shaft material is 120588 =7850 kgm3 and the Poisson ratio of the shaft is V = 03Table 1 lists the physical parameters of five gears Five shaftsunbalance and location calculated by ISO-1940-12003 [20]is listed in Table 2 and tilting pad bearing parameters of fiveshafts are presented in the Table 3 All the other data for thesystem are shown in the Figure 7

Table 1 Physical parameters of five gears

Input Middle Output 1 Output 2 Output 3Power (kW) 15600 15600 4600 6500 4500Pitch diameter (mm) 719 1984 264 221 141Tooth width (mm) 224 152 224 224 224Location 40∘ 0∘ 270∘ 90∘ 180∘

Normal modulus 5Pressure angle 20∘

Helix angle 12∘

Table 2 Five shafts unbalance and location

Input Middle Output 1 Output 2 Output 3Mass (kg) 109869 429915 38329 35903 14326Unbalance(gmm) 4395 64487 958 359 359 72 72

Locationnode 2 38 88 89 133 144 174

Table 3 Bearing parameters of five shafts

Input Middle Output 1 Output 2 Output 3Bearingdiameter(mm)

180 250 100 110 80

Bearingwidth (mm) 176 150 90 98 72

Bearingradialclearance(mm)

006 005 009 0116 0096

Pad number 4 2 5 5 5Rated speed(rpm) 4087 1481 11131 13297 20841

When running at the rated speeds integrally gearedcompressor needs to adjust the load according to the practicalrequirements Since the interaction of the geared rotor-bearing system the workload is associated with the bearingdynamic coefficients axial force and torque on the shaft andgear meshing stiffness In the integrally geared compressorthe input shaft and the middle shaft bearing the workloadare generated by all the output shafts and usually run belowthe first critical speed Therefore resonance is difficult toproduce with the workload adjustment when the input shaftand middle shaft are running at the rated speed On thecontrary output shafts have rated speeds higher than thecritical speed When running at the rated speeds they mayproduce resonance with the workload adjustment and thecritical load phenomenon occurs This vibration problemis investigated by linear unbalance response analysis of thegeared rotor-bearing system with varying workloads

In order to construct the geared rotor-bearing systemmodel we utilize the meshing stiffness of the gear pair toconnect the five single rotor-bearing systems The motionof the gear pair is determined by the gear mesh stiffness

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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2 Shock and Vibration

geared rotor-bearing systems have also been widely explored[15ndash17]

The complicated dynamic behaviors including linear andnonlinear characteristics of geared rotor-bearing systemshave been extensively investigated However the dynamiccharacteristics of geared rotor-bearing systems consideringvaryingworkload have been rarely explored Integrally gearedcompressors likely cross numerous critical speeds duringcompressor start up When running at rated speeds inte-grally geared compressors produce resonance with adjustedworkload and a critical load phenomenon occurs Gao[18] discovered and successfully resolved a DH integrallygeared compressor failure caused by critical load Thereforeour study mainly aims to examine the effect of differentworkloads on the dynamic characteristics of a geared rotor-bearing system and to investigate the failure mechanism ofthe critical load

In this paper a general finite element approach of gearedrotor-bearing system is presented in Section 2 The dynamicmodel of geared rotor-bearing system covers the effects ofbearing flexibility and damping axial force and torque andgear meshing A five-shaft integrally geared compressor ismodeled and discussed in Section 3 The bearing dynamiccoefficients axial force and torque and gearmeshing stiffnessare changed as workload varies because of the interaction inthe geared rotor-bearing system The dynamic responses ofthe system under different parameters are also analyzed Ourresults indicate that the bearing dynamic coefficients and gearmeshing stiffness significantly influence the responses of thegeared rotor-bearing system but the axial force and torqueslightly influence such responses The resonance peaks andtheir phase of the critical load are obtained from the dynamicresponses affected by the bearing dynamic coefficients axialforce and torque and gear meshing stiffness

2 Dynamic Modeling for GearedRotor-Bearing System

In this section the FE dynamic equation of the geared rotor-bearing system influenced by the flexibility and dampingof bearings axial force axial torque and gear meshing ispresented The assembly method of the coupling systemequation is also investigated

21 Rotor-Bearing System Dynamic Equation The coordi-nates defining bending as the two planes are shown inFigure 1 The local coordinate of the finite rotor element is119902119890 = [119906119894 V119894 120579119894 120595119894 119906119895 V119895 120579119895 120595119895] The dynamic equation of thefinite rotor element is as follows

(119872119890119879 +119872119890119877) 119890 minus Ω119866119890119890 + 119870119890119861119902119890 = 119876119890 (1)

where 119872119890119879 and 119872119890119877 are the element mass matrices 119870119890119861 isthe element bending stiffness matrix 119866119890 is the elementgyroscopicmatrixΩ is the spin speed119876119890 is the external forcevector of the rotor element The matrices in (1) are defined inAppendix A

The coupling of the input shaft the impellers of outputshafts and the gear of the middle shaft are regarded as a rigid

disk (with gyroscopic effect) When the local coordinates aredefined as 119902119889 = [119906 V 120579 120595] the dynamic equation of the rigiddisk is as follows

119872119889119889 minus Ω119866119889119889 = 119876119889 (2)

where Ω is the spin speed 119876119889 is the external force vector ofthe rigid discs119872119889 and119866119889 are themassmatrix and gyroscopicmatrix for the disk respectively and defined as follows

119872119889 =[[[[[[

119898119889 0 0 00 119898119889 0 00 0 119868119889 00 0 0 119868119889

]]]]]]

119866119889 =[[[[[[

0 0 0 00 0 0 00 0 0 1198681199010 0 minus119868119901 0

]]]]]]

(3)

The bearings utilized in this paper are limited to thosewhich obey the governing equations of the form

119862119887119887 + 119870119887119902119887 = 119876119887 (4)

in the local coordinates where119876119887 is the bearing external forcevector 119870119887 = [ 119896119906119906 119896119906V119896V119906 119896VV

] is the stiffness matrix of bearing and119862119887 = [ 119888119906119906 119888119906V119888V119906 119888VV ] is the damping matrix of bearing

22 Incremental Stiffness due to Axial Force and Axial TorqueIn the integrally geared compressor the mechanical powerdeveloped by the turbine is transmitted through the inputshaft to the output shaft impellers where gas is pressurizedThe purpose of the geared rotor-bearing system usuallydepends fundamentally on this transmission of torque Eachshaft of the compressor carries some torque about the axisof rotation The helical gear meshing is generally usedfor integrally geared compressors to ensure the stability oftransmission The mechanical power of integrally gearedcompressors contributes to not only the axial torque but alsothe axial force because of the effect of helical gear meshingThe axial force along the undeformed axis of the rotor occursin each shaft of the coupling system Nelson andMcVaugh [2]developed the equation of motion for a rotating finite shaftelement including the effect of axial force The result is astiffening effect if the shaft is in tension or a softening effect

Shock and Vibration 3

z

ui

120595i

uj

120595j

i

120579i

j

120579j

z

x-z plane

y-z plane

Figure 1 The local coordinates in the bending planes

if the force is compressive The lateral motion of an Euler-Bernoulli beam element of the axial force is as follows

119870119890119860

= 11986511988630119897119890

[[[[[[[[[[[[[[[[[[

360 36 sym

0 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36 3119897119890 0 0 360 minus3119897119890 minus1198972119890 0 0 3119897119890 411989721198903119897 0 0 minus1198972119890 minus3119897119890 0 0 4119897119890

]]]]]]]]]]]]]]]]]]

(5)

where 119865119886 is the axial tensile force within the elementThe axial torque within a shaft may affect the lateral

behavior which is similar to the effect caused by the axialforce in a shaft Relatively few papers or texts on rotordynamics consider this coupling although Zorzi and Nelson[4] derive the contribution to the stiffness for the lateralmotion of an Euler-Bernoulli beam element as follows

119870119890119879 = 1205911198902119897119890

[[[[[[[[[[[[[[[[[[

0 0 2 0 0 0 minus2 00 0 0 2 0 0 0 minus22 0 0 minus119897119890 minus2 0 0 1198971198900 2 119897119890 0 0 minus2 minus119897119890 00 0 minus2 0 0 0 2 00 0 0 minus2 0 0 0 2minus2 0 0 minus119897119890 2 0 0 1198971198900 minus2 119897119890 0 0 2 minus119897119890 0

]]]]]]]]]]]]]]]]]]

(6)

where 120591119890 is the transmitted torque within the element

23 Gear Meshing Stiffness Connecting the shafts throughthe helical gear in the geared rotor-bearing system oftencauses couple vibration Helical gear meshing stiffness isinvestigated on the basis of a previous study [6] The rela-tionship between contact force and displacements of a gearpair is determined by representing the teeth in contact withan equivalent stiffness along the pressure line as shown inFigure 2 The rotations 120579 and 120595 are small in a gear pair assuch we can disregard these terms Along the pressure linegear119894 moves as follows

119909119894119892 = 119906119894 sin120572 cos120573 + V119894 cos120572 cos120573 + 119903119894120601119894 cos120573 (7)

and gear119895 moves as follows

119909119895119892 = 119906119895 sin120572 cos120573 + V119895 cos120572 cos120573 + 119903119895120601119895 cos120573 (8)

where 119903119894 and 119903119895 are the base circle radii of the two gears 120572 isthe pressure angle 120573 is helix angle 120601119894 and 120601119895 are the torsionaldisplacements of the gears about the 119911-axil 119906119894 and 119906119895 are thedisplacement of gear119894 and gear119895 in the 119909 direction V119894 and V119895are the displacement of gear119894 and gear119895 in the 119910 direction

Therefore the relative displacement at the gear mesh is asfollows

119909119892 = (119906119895 minus 119906119894) sin120572 cos120573 + (V119895 minus V119894) cos120572 cos120573minus (119903119894120601119894 + 119903119895120601119895) cos120573 = 119873119892119902119892

(9)

where

119873119892 = [minus sin120572 cos120573 minus cos120572 sin120573 0 0 sin120572 cos120573 cos120572 cos120573 0 0 minus119903119894 cos120573 minus119903119895 cos120573]

119902119892 = [119906119894 V119894 120579119894 120595119894 119906119895 V119895 120579119895 120595119895 120601119894 120601119895] (10)

The strain energy within the gear mesh is as follows

119880119892 = 121198961198921199092119892 = 1

2119896119892119902119879119892119873119879119892119873119892119902119892 (11)

where 119896119892 is the gear mesh stiffness Thus the stiffness matrixof gear mesh is as follows

119870119892 = 119896119892119873119879119892119873119892 (12)

The matrix of119870119892 is defined in Appendix B

4 Shock and Vibration

24 The Dynamic Equation of Geared Rotor-Bearing SystemEquations (1) (2) (5) (6) and (12) can be combined forthe geared coupled system The dynamic equation of gearedrotor-bearing system is as follows

119872119904119904 + (119862119904119887 minus Ω119866119904) 119904+ (119870119904119861 + 119870119904119887 + 119870119904119892 minus 119870119904119860 minus 119870119904119879) 119902119904 = 119876119904 (13)

where 119872119904 is the mass matrix 119862119904119887 is the damping matrix ofbearing Ω is the spin speed 119866119904 is the gyroscopic matrix119870119904119861 119870119904119887 119870119904119892 119870119904119860 and 119870119904119879 are the stiffness matrix of the rotorbearing gear axial load and axial torque respectively 119876119904 isthe external force vector 119902119904 is the displacement vector whichis defined as follows119902119904 = [1199061 V1 1205791 1205951 1206011 1199062 V2 1205792 1205952 1206012 119906119899 V119899 120579119899 120595119899

120601119899] (14)

25 Assembly of the Coupling System Equation In order toconstruct an entire geared rotor-bearing system equationan assembly method of the single rotor finite element (FE)models of the shafts bearings and disks and the gear pairFE models is stated as follows The single rotor FE modelassembly method is detailed in the book by Vollan andKomzsik [19] Figure 3 shows such a simple geared systemwhich is used as an example to illustrate the assemblymethod of the entire system equation The nodes on eachgear are connected by a stiffness matrix In a geared systemthe element matrices are inserted into the system matricesin the positions determined by the positions of the localelement coordinates in the global vectorThis is demonstrateddiagrammatically in Figure 4 for the example of Figure 3 inwhich the nonzero degrees of freedom are shadedThe blocksin shades of light gray and dark gray denote the positionscorresponding to Rotor 1 and Rotor 2 respectively The blacksquares denote the stiffness matrix for the connection of thegear meshing which is split into four blocks that slot into theposition of Node 3 and Node 6 in the entire system equationThen the entire assembled equation of a geared system isimplemented as shown in Figure 4

26 Validation of Dynamic Model A 600 kW turbo-chillerrotor-bearing system with a bull-pinion speed-increasinggear is applied to validate the correctness of the modelproposed in this paper The dynamic model of the turbo-chiller rotor-bearing system is shown in Figure 5 and thesystem parameters are based on a previous study [9] Witha test unbalance of 119880 = 197 gsdotmm attached to the impellerthe coupled unbalance responses at the driver shaft calculatedby the proposed method are shown in Figure 6 Our results(Figure 6) are consistent with those described in a previousstudy [9]

3 Results and Discussions

Applying the proposed method we perform an unbalanceresponse analysis with a five-shaft geared rotor-bearing sys-tem of an integrally geared compressor (Figure 7) The input

ui

i

uj

j

i

i

j

j

i

kgjGear

Gear zi

zj

Ti

Tj

Oi

Oj

i

j

Figure 2 Gear meshing system model

Node 1 Node 2

Node 3

Node 4 Node 5

Node 6

Node 7 Node 8

Rotor 1

Rotor 2

Figure 3 Rotors coupled by a gear

Rotor 1 Rotor 2

Figure 4 Structure of the entire assembled equation for a gearedsystem

Shock and Vibration 5

D1 D2 D3 D4 G1

D5 D6D7 D8B1 B2

B3 B4

Driver shaft

Driven shaft

Figure 5 Dynamic model of the turbo-chiller rotor-bearing systemin [9]

At bull gear (this paper)At bull gear ([9])At motor side bearing (this paper)At motor side bearing ([9])

1500 2000 2500 3000 35001000Rotating speed (rpm)

Resp

onse

(m

pea

k to

pea

k)

0

001

002

003

004

005

006

Figure 6 Unbalance responses comparing with [9]

shaft and three output shafts (O1 O2 and O3) are coupledtogether by the transmission gear of the middle shaft in thiscoupling system Each shaft is a general rotor-bearing systemsupported by oil film bearings The impellers are fixed on theoutput shafts to compress the gas

In the five-shaft geared rotor-bearing system each shafthas the same material properties Youngrsquos modulus of shaftis 119864 = 2059 times 1011Nm2 shear modulus of shaft is 119866 =7919 times 1010Nm2 the density of the shaft material is 120588 =7850 kgm3 and the Poisson ratio of the shaft is V = 03Table 1 lists the physical parameters of five gears Five shaftsunbalance and location calculated by ISO-1940-12003 [20]is listed in Table 2 and tilting pad bearing parameters of fiveshafts are presented in the Table 3 All the other data for thesystem are shown in the Figure 7

Table 1 Physical parameters of five gears

Input Middle Output 1 Output 2 Output 3Power (kW) 15600 15600 4600 6500 4500Pitch diameter (mm) 719 1984 264 221 141Tooth width (mm) 224 152 224 224 224Location 40∘ 0∘ 270∘ 90∘ 180∘

Normal modulus 5Pressure angle 20∘

Helix angle 12∘

Table 2 Five shafts unbalance and location

Input Middle Output 1 Output 2 Output 3Mass (kg) 109869 429915 38329 35903 14326Unbalance(gmm) 4395 64487 958 359 359 72 72

Locationnode 2 38 88 89 133 144 174

Table 3 Bearing parameters of five shafts

Input Middle Output 1 Output 2 Output 3Bearingdiameter(mm)

180 250 100 110 80

Bearingwidth (mm) 176 150 90 98 72

Bearingradialclearance(mm)

006 005 009 0116 0096

Pad number 4 2 5 5 5Rated speed(rpm) 4087 1481 11131 13297 20841

When running at the rated speeds integrally gearedcompressor needs to adjust the load according to the practicalrequirements Since the interaction of the geared rotor-bearing system the workload is associated with the bearingdynamic coefficients axial force and torque on the shaft andgear meshing stiffness In the integrally geared compressorthe input shaft and the middle shaft bearing the workloadare generated by all the output shafts and usually run belowthe first critical speed Therefore resonance is difficult toproduce with the workload adjustment when the input shaftand middle shaft are running at the rated speed On thecontrary output shafts have rated speeds higher than thecritical speed When running at the rated speeds they mayproduce resonance with the workload adjustment and thecritical load phenomenon occurs This vibration problemis investigated by linear unbalance response analysis of thegeared rotor-bearing system with varying workloads

In order to construct the geared rotor-bearing systemmodel we utilize the meshing stiffness of the gear pair toconnect the five single rotor-bearing systems The motionof the gear pair is determined by the gear mesh stiffness

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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Shock and Vibration

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

z

ui

120595i

uj

120595j

i

120579i

j

120579j

z

x-z plane

y-z plane

Figure 1 The local coordinates in the bending planes

if the force is compressive The lateral motion of an Euler-Bernoulli beam element of the axial force is as follows

119870119890119860

= 11986511988630119897119890

[[[[[[[[[[[[[[[[[[

360 36 sym

0 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36 3119897119890 0 0 360 minus3119897119890 minus1198972119890 0 0 3119897119890 411989721198903119897 0 0 minus1198972119890 minus3119897119890 0 0 4119897119890

]]]]]]]]]]]]]]]]]]

(5)

where 119865119886 is the axial tensile force within the elementThe axial torque within a shaft may affect the lateral

behavior which is similar to the effect caused by the axialforce in a shaft Relatively few papers or texts on rotordynamics consider this coupling although Zorzi and Nelson[4] derive the contribution to the stiffness for the lateralmotion of an Euler-Bernoulli beam element as follows

119870119890119879 = 1205911198902119897119890

[[[[[[[[[[[[[[[[[[

0 0 2 0 0 0 minus2 00 0 0 2 0 0 0 minus22 0 0 minus119897119890 minus2 0 0 1198971198900 2 119897119890 0 0 minus2 minus119897119890 00 0 minus2 0 0 0 2 00 0 0 minus2 0 0 0 2minus2 0 0 minus119897119890 2 0 0 1198971198900 minus2 119897119890 0 0 2 minus119897119890 0

]]]]]]]]]]]]]]]]]]

(6)

where 120591119890 is the transmitted torque within the element

23 Gear Meshing Stiffness Connecting the shafts throughthe helical gear in the geared rotor-bearing system oftencauses couple vibration Helical gear meshing stiffness isinvestigated on the basis of a previous study [6] The rela-tionship between contact force and displacements of a gearpair is determined by representing the teeth in contact withan equivalent stiffness along the pressure line as shown inFigure 2 The rotations 120579 and 120595 are small in a gear pair assuch we can disregard these terms Along the pressure linegear119894 moves as follows

119909119894119892 = 119906119894 sin120572 cos120573 + V119894 cos120572 cos120573 + 119903119894120601119894 cos120573 (7)

and gear119895 moves as follows

119909119895119892 = 119906119895 sin120572 cos120573 + V119895 cos120572 cos120573 + 119903119895120601119895 cos120573 (8)

where 119903119894 and 119903119895 are the base circle radii of the two gears 120572 isthe pressure angle 120573 is helix angle 120601119894 and 120601119895 are the torsionaldisplacements of the gears about the 119911-axil 119906119894 and 119906119895 are thedisplacement of gear119894 and gear119895 in the 119909 direction V119894 and V119895are the displacement of gear119894 and gear119895 in the 119910 direction

Therefore the relative displacement at the gear mesh is asfollows

119909119892 = (119906119895 minus 119906119894) sin120572 cos120573 + (V119895 minus V119894) cos120572 cos120573minus (119903119894120601119894 + 119903119895120601119895) cos120573 = 119873119892119902119892

(9)

where

119873119892 = [minus sin120572 cos120573 minus cos120572 sin120573 0 0 sin120572 cos120573 cos120572 cos120573 0 0 minus119903119894 cos120573 minus119903119895 cos120573]

119902119892 = [119906119894 V119894 120579119894 120595119894 119906119895 V119895 120579119895 120595119895 120601119894 120601119895] (10)

The strain energy within the gear mesh is as follows

119880119892 = 121198961198921199092119892 = 1

2119896119892119902119879119892119873119879119892119873119892119902119892 (11)

where 119896119892 is the gear mesh stiffness Thus the stiffness matrixof gear mesh is as follows

119870119892 = 119896119892119873119879119892119873119892 (12)

The matrix of119870119892 is defined in Appendix B

4 Shock and Vibration

24 The Dynamic Equation of Geared Rotor-Bearing SystemEquations (1) (2) (5) (6) and (12) can be combined forthe geared coupled system The dynamic equation of gearedrotor-bearing system is as follows

119872119904119904 + (119862119904119887 minus Ω119866119904) 119904+ (119870119904119861 + 119870119904119887 + 119870119904119892 minus 119870119904119860 minus 119870119904119879) 119902119904 = 119876119904 (13)

where 119872119904 is the mass matrix 119862119904119887 is the damping matrix ofbearing Ω is the spin speed 119866119904 is the gyroscopic matrix119870119904119861 119870119904119887 119870119904119892 119870119904119860 and 119870119904119879 are the stiffness matrix of the rotorbearing gear axial load and axial torque respectively 119876119904 isthe external force vector 119902119904 is the displacement vector whichis defined as follows119902119904 = [1199061 V1 1205791 1205951 1206011 1199062 V2 1205792 1205952 1206012 119906119899 V119899 120579119899 120595119899

120601119899] (14)

25 Assembly of the Coupling System Equation In order toconstruct an entire geared rotor-bearing system equationan assembly method of the single rotor finite element (FE)models of the shafts bearings and disks and the gear pairFE models is stated as follows The single rotor FE modelassembly method is detailed in the book by Vollan andKomzsik [19] Figure 3 shows such a simple geared systemwhich is used as an example to illustrate the assemblymethod of the entire system equation The nodes on eachgear are connected by a stiffness matrix In a geared systemthe element matrices are inserted into the system matricesin the positions determined by the positions of the localelement coordinates in the global vectorThis is demonstrateddiagrammatically in Figure 4 for the example of Figure 3 inwhich the nonzero degrees of freedom are shadedThe blocksin shades of light gray and dark gray denote the positionscorresponding to Rotor 1 and Rotor 2 respectively The blacksquares denote the stiffness matrix for the connection of thegear meshing which is split into four blocks that slot into theposition of Node 3 and Node 6 in the entire system equationThen the entire assembled equation of a geared system isimplemented as shown in Figure 4

26 Validation of Dynamic Model A 600 kW turbo-chillerrotor-bearing system with a bull-pinion speed-increasinggear is applied to validate the correctness of the modelproposed in this paper The dynamic model of the turbo-chiller rotor-bearing system is shown in Figure 5 and thesystem parameters are based on a previous study [9] Witha test unbalance of 119880 = 197 gsdotmm attached to the impellerthe coupled unbalance responses at the driver shaft calculatedby the proposed method are shown in Figure 6 Our results(Figure 6) are consistent with those described in a previousstudy [9]

3 Results and Discussions

Applying the proposed method we perform an unbalanceresponse analysis with a five-shaft geared rotor-bearing sys-tem of an integrally geared compressor (Figure 7) The input

ui

i

uj

j

i

i

j

j

i

kgjGear

Gear zi

zj

Ti

Tj

Oi

Oj

i

j

Figure 2 Gear meshing system model

Node 1 Node 2

Node 3

Node 4 Node 5

Node 6

Node 7 Node 8

Rotor 1

Rotor 2

Figure 3 Rotors coupled by a gear

Rotor 1 Rotor 2

Figure 4 Structure of the entire assembled equation for a gearedsystem

Shock and Vibration 5

D1 D2 D3 D4 G1

D5 D6D7 D8B1 B2

B3 B4

Driver shaft

Driven shaft

Figure 5 Dynamic model of the turbo-chiller rotor-bearing systemin [9]

At bull gear (this paper)At bull gear ([9])At motor side bearing (this paper)At motor side bearing ([9])

1500 2000 2500 3000 35001000Rotating speed (rpm)

Resp

onse

(m

pea

k to

pea

k)

0

001

002

003

004

005

006

Figure 6 Unbalance responses comparing with [9]

shaft and three output shafts (O1 O2 and O3) are coupledtogether by the transmission gear of the middle shaft in thiscoupling system Each shaft is a general rotor-bearing systemsupported by oil film bearings The impellers are fixed on theoutput shafts to compress the gas

In the five-shaft geared rotor-bearing system each shafthas the same material properties Youngrsquos modulus of shaftis 119864 = 2059 times 1011Nm2 shear modulus of shaft is 119866 =7919 times 1010Nm2 the density of the shaft material is 120588 =7850 kgm3 and the Poisson ratio of the shaft is V = 03Table 1 lists the physical parameters of five gears Five shaftsunbalance and location calculated by ISO-1940-12003 [20]is listed in Table 2 and tilting pad bearing parameters of fiveshafts are presented in the Table 3 All the other data for thesystem are shown in the Figure 7

Table 1 Physical parameters of five gears

Input Middle Output 1 Output 2 Output 3Power (kW) 15600 15600 4600 6500 4500Pitch diameter (mm) 719 1984 264 221 141Tooth width (mm) 224 152 224 224 224Location 40∘ 0∘ 270∘ 90∘ 180∘

Normal modulus 5Pressure angle 20∘

Helix angle 12∘

Table 2 Five shafts unbalance and location

Input Middle Output 1 Output 2 Output 3Mass (kg) 109869 429915 38329 35903 14326Unbalance(gmm) 4395 64487 958 359 359 72 72

Locationnode 2 38 88 89 133 144 174

Table 3 Bearing parameters of five shafts

Input Middle Output 1 Output 2 Output 3Bearingdiameter(mm)

180 250 100 110 80

Bearingwidth (mm) 176 150 90 98 72

Bearingradialclearance(mm)

006 005 009 0116 0096

Pad number 4 2 5 5 5Rated speed(rpm) 4087 1481 11131 13297 20841

When running at the rated speeds integrally gearedcompressor needs to adjust the load according to the practicalrequirements Since the interaction of the geared rotor-bearing system the workload is associated with the bearingdynamic coefficients axial force and torque on the shaft andgear meshing stiffness In the integrally geared compressorthe input shaft and the middle shaft bearing the workloadare generated by all the output shafts and usually run belowthe first critical speed Therefore resonance is difficult toproduce with the workload adjustment when the input shaftand middle shaft are running at the rated speed On thecontrary output shafts have rated speeds higher than thecritical speed When running at the rated speeds they mayproduce resonance with the workload adjustment and thecritical load phenomenon occurs This vibration problemis investigated by linear unbalance response analysis of thegeared rotor-bearing system with varying workloads

In order to construct the geared rotor-bearing systemmodel we utilize the meshing stiffness of the gear pair toconnect the five single rotor-bearing systems The motionof the gear pair is determined by the gear mesh stiffness

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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International Journal of

4 Shock and Vibration

24 The Dynamic Equation of Geared Rotor-Bearing SystemEquations (1) (2) (5) (6) and (12) can be combined forthe geared coupled system The dynamic equation of gearedrotor-bearing system is as follows

119872119904119904 + (119862119904119887 minus Ω119866119904) 119904+ (119870119904119861 + 119870119904119887 + 119870119904119892 minus 119870119904119860 minus 119870119904119879) 119902119904 = 119876119904 (13)

where 119872119904 is the mass matrix 119862119904119887 is the damping matrix ofbearing Ω is the spin speed 119866119904 is the gyroscopic matrix119870119904119861 119870119904119887 119870119904119892 119870119904119860 and 119870119904119879 are the stiffness matrix of the rotorbearing gear axial load and axial torque respectively 119876119904 isthe external force vector 119902119904 is the displacement vector whichis defined as follows119902119904 = [1199061 V1 1205791 1205951 1206011 1199062 V2 1205792 1205952 1206012 119906119899 V119899 120579119899 120595119899

120601119899] (14)

25 Assembly of the Coupling System Equation In order toconstruct an entire geared rotor-bearing system equationan assembly method of the single rotor finite element (FE)models of the shafts bearings and disks and the gear pairFE models is stated as follows The single rotor FE modelassembly method is detailed in the book by Vollan andKomzsik [19] Figure 3 shows such a simple geared systemwhich is used as an example to illustrate the assemblymethod of the entire system equation The nodes on eachgear are connected by a stiffness matrix In a geared systemthe element matrices are inserted into the system matricesin the positions determined by the positions of the localelement coordinates in the global vectorThis is demonstrateddiagrammatically in Figure 4 for the example of Figure 3 inwhich the nonzero degrees of freedom are shadedThe blocksin shades of light gray and dark gray denote the positionscorresponding to Rotor 1 and Rotor 2 respectively The blacksquares denote the stiffness matrix for the connection of thegear meshing which is split into four blocks that slot into theposition of Node 3 and Node 6 in the entire system equationThen the entire assembled equation of a geared system isimplemented as shown in Figure 4

26 Validation of Dynamic Model A 600 kW turbo-chillerrotor-bearing system with a bull-pinion speed-increasinggear is applied to validate the correctness of the modelproposed in this paper The dynamic model of the turbo-chiller rotor-bearing system is shown in Figure 5 and thesystem parameters are based on a previous study [9] Witha test unbalance of 119880 = 197 gsdotmm attached to the impellerthe coupled unbalance responses at the driver shaft calculatedby the proposed method are shown in Figure 6 Our results(Figure 6) are consistent with those described in a previousstudy [9]

3 Results and Discussions

Applying the proposed method we perform an unbalanceresponse analysis with a five-shaft geared rotor-bearing sys-tem of an integrally geared compressor (Figure 7) The input

ui

i

uj

j

i

i

j

j

i

kgjGear

Gear zi

zj

Ti

Tj

Oi

Oj

i

j

Figure 2 Gear meshing system model

Node 1 Node 2

Node 3

Node 4 Node 5

Node 6

Node 7 Node 8

Rotor 1

Rotor 2

Figure 3 Rotors coupled by a gear

Rotor 1 Rotor 2

Figure 4 Structure of the entire assembled equation for a gearedsystem

Shock and Vibration 5

D1 D2 D3 D4 G1

D5 D6D7 D8B1 B2

B3 B4

Driver shaft

Driven shaft

Figure 5 Dynamic model of the turbo-chiller rotor-bearing systemin [9]

At bull gear (this paper)At bull gear ([9])At motor side bearing (this paper)At motor side bearing ([9])

1500 2000 2500 3000 35001000Rotating speed (rpm)

Resp

onse

(m

pea

k to

pea

k)

0

001

002

003

004

005

006

Figure 6 Unbalance responses comparing with [9]

shaft and three output shafts (O1 O2 and O3) are coupledtogether by the transmission gear of the middle shaft in thiscoupling system Each shaft is a general rotor-bearing systemsupported by oil film bearings The impellers are fixed on theoutput shafts to compress the gas

In the five-shaft geared rotor-bearing system each shafthas the same material properties Youngrsquos modulus of shaftis 119864 = 2059 times 1011Nm2 shear modulus of shaft is 119866 =7919 times 1010Nm2 the density of the shaft material is 120588 =7850 kgm3 and the Poisson ratio of the shaft is V = 03Table 1 lists the physical parameters of five gears Five shaftsunbalance and location calculated by ISO-1940-12003 [20]is listed in Table 2 and tilting pad bearing parameters of fiveshafts are presented in the Table 3 All the other data for thesystem are shown in the Figure 7

Table 1 Physical parameters of five gears

Input Middle Output 1 Output 2 Output 3Power (kW) 15600 15600 4600 6500 4500Pitch diameter (mm) 719 1984 264 221 141Tooth width (mm) 224 152 224 224 224Location 40∘ 0∘ 270∘ 90∘ 180∘

Normal modulus 5Pressure angle 20∘

Helix angle 12∘

Table 2 Five shafts unbalance and location

Input Middle Output 1 Output 2 Output 3Mass (kg) 109869 429915 38329 35903 14326Unbalance(gmm) 4395 64487 958 359 359 72 72

Locationnode 2 38 88 89 133 144 174

Table 3 Bearing parameters of five shafts

Input Middle Output 1 Output 2 Output 3Bearingdiameter(mm)

180 250 100 110 80

Bearingwidth (mm) 176 150 90 98 72

Bearingradialclearance(mm)

006 005 009 0116 0096

Pad number 4 2 5 5 5Rated speed(rpm) 4087 1481 11131 13297 20841

When running at the rated speeds integrally gearedcompressor needs to adjust the load according to the practicalrequirements Since the interaction of the geared rotor-bearing system the workload is associated with the bearingdynamic coefficients axial force and torque on the shaft andgear meshing stiffness In the integrally geared compressorthe input shaft and the middle shaft bearing the workloadare generated by all the output shafts and usually run belowthe first critical speed Therefore resonance is difficult toproduce with the workload adjustment when the input shaftand middle shaft are running at the rated speed On thecontrary output shafts have rated speeds higher than thecritical speed When running at the rated speeds they mayproduce resonance with the workload adjustment and thecritical load phenomenon occurs This vibration problemis investigated by linear unbalance response analysis of thegeared rotor-bearing system with varying workloads

In order to construct the geared rotor-bearing systemmodel we utilize the meshing stiffness of the gear pair toconnect the five single rotor-bearing systems The motionof the gear pair is determined by the gear mesh stiffness

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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International Journal of

Shock and Vibration 5

D1 D2 D3 D4 G1

D5 D6D7 D8B1 B2

B3 B4

Driver shaft

Driven shaft

Figure 5 Dynamic model of the turbo-chiller rotor-bearing systemin [9]

At bull gear (this paper)At bull gear ([9])At motor side bearing (this paper)At motor side bearing ([9])

1500 2000 2500 3000 35001000Rotating speed (rpm)

Resp

onse

(m

pea

k to

pea

k)

0

001

002

003

004

005

006

Figure 6 Unbalance responses comparing with [9]

shaft and three output shafts (O1 O2 and O3) are coupledtogether by the transmission gear of the middle shaft in thiscoupling system Each shaft is a general rotor-bearing systemsupported by oil film bearings The impellers are fixed on theoutput shafts to compress the gas

In the five-shaft geared rotor-bearing system each shafthas the same material properties Youngrsquos modulus of shaftis 119864 = 2059 times 1011Nm2 shear modulus of shaft is 119866 =7919 times 1010Nm2 the density of the shaft material is 120588 =7850 kgm3 and the Poisson ratio of the shaft is V = 03Table 1 lists the physical parameters of five gears Five shaftsunbalance and location calculated by ISO-1940-12003 [20]is listed in Table 2 and tilting pad bearing parameters of fiveshafts are presented in the Table 3 All the other data for thesystem are shown in the Figure 7

Table 1 Physical parameters of five gears

Input Middle Output 1 Output 2 Output 3Power (kW) 15600 15600 4600 6500 4500Pitch diameter (mm) 719 1984 264 221 141Tooth width (mm) 224 152 224 224 224Location 40∘ 0∘ 270∘ 90∘ 180∘

Normal modulus 5Pressure angle 20∘

Helix angle 12∘

Table 2 Five shafts unbalance and location

Input Middle Output 1 Output 2 Output 3Mass (kg) 109869 429915 38329 35903 14326Unbalance(gmm) 4395 64487 958 359 359 72 72

Locationnode 2 38 88 89 133 144 174

Table 3 Bearing parameters of five shafts

Input Middle Output 1 Output 2 Output 3Bearingdiameter(mm)

180 250 100 110 80

Bearingwidth (mm) 176 150 90 98 72

Bearingradialclearance(mm)

006 005 009 0116 0096

Pad number 4 2 5 5 5Rated speed(rpm) 4087 1481 11131 13297 20841

When running at the rated speeds integrally gearedcompressor needs to adjust the load according to the practicalrequirements Since the interaction of the geared rotor-bearing system the workload is associated with the bearingdynamic coefficients axial force and torque on the shaft andgear meshing stiffness In the integrally geared compressorthe input shaft and the middle shaft bearing the workloadare generated by all the output shafts and usually run belowthe first critical speed Therefore resonance is difficult toproduce with the workload adjustment when the input shaftand middle shaft are running at the rated speed On thecontrary output shafts have rated speeds higher than thecritical speed When running at the rated speeds they mayproduce resonance with the workload adjustment and thecritical load phenomenon occurs This vibration problemis investigated by linear unbalance response analysis of thegeared rotor-bearing system with varying workloads

In order to construct the geared rotor-bearing systemmodel we utilize the meshing stiffness of the gear pair toconnect the five single rotor-bearing systems The motionof the gear pair is determined by the gear mesh stiffness

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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Shock and Vibration

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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

6 Shock and Vibration

Output shaft 1

Output shaft 2

Output shaft 3

Input shaft

Middle shaft

B-node 79B-node 57

B-node 32

B-node 100 B-node 122

B-node 19B-node 9D-node 2

D-node 89 D-node 133

D-node 88

D-node 174B-node 164B-node 144D-node 174

B-node 45

m = 1746kg

ID = 00655kgm2

IP = 0104kgm2

m = 5823kg

ID = 0544kgm2

IP = 0883kgm2

m = 1220kg

ID = 1055kgm2

IP = 211kgm2

m = 1463kg

ID = 00486kgm2

IP = 00731kgm2

m = 1127kg

ID = 1661kgm2

IP = 2763kgm2

m = 5523kg

ID = 0485kgm2

IP = 0791kgm2

Figure 7 Five-shaft geared rotor-bearing system of integrally geared compressors

(see Section 23) Using the proposed model to calculatethe linear unbalance response of the geared rotor-bearingsystem the resonance speed of the three output shafts underthe interaction of the gears can be obtained The vibrationproblem of geared rotor-bearing system is discussed andanalyzed in detail based on the three-dimensional map oflinear unbalance response with varying workloads

In this section we discussed the relationship betweenthe workload and the bearing dynamic coefficients axialforce and torque and gear meshing stiffness obtainedrespectively linear unbalance response under the effect ofbearing dynamic coefficients axial force and torque and gearmeshing stiffness by using the dynamic modeling method inSection 2 and analyzed the influence of different factors onthe critical speed in the process of workload variation Thenby analyzing the unbalance response under the influence ofall factors the critical load of the geared rotor-bearing systemis verified

31 Unbalance Response Analysis of the Geared Rotor-BearingSystem considering VaryingWorkload According to the forceanalysis of typical helical gears (Figure 8) we can obtain

119865119905 = 2119879119889

119865119903 = 119865119905 tan120572cos120573

119865119886 = 119865119905 tan120573

(15)

b

b998400 120573b

a998400

a

Fn

c

Fr

120572

120573F

120572t

Ft

P

l

d

T

Fa

e

Figure 8 Force analysis of helical gears

where 119865119905 is the circumferential force 119865119903 is the radial force 119865119886is the axial force 119879 is the torque 119889 is the pitch diameter 120572 isthe pressure angle 120573 is the helix angle

When running at the rated speed the change of theworkload has directly led to torque variant based on theformula which is

119879 = 9550119875119899 (16)

Shock and Vibration 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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 7

20 40 60 80 1000Load ()

108

109

1010

K (N

m)

O1-Kxx

O1-Kyy

O2-Kxx

O2-Kyy

O3-Kxx

O3-Kyy

(a)

O1-CxxO1-CyyO2-Cxx

O2-CyyO3-CxxO3-Cyy

105

106

107

C (N

sm

)

20 40 60 80 1000Load ()

(b)

Figure 9 Effect of increasing workload on the tilting pad bearings of three output shafts (a) stiffness coefficients (b) damping coefficients

where 119879 is the torque 119875 is the workload 119899 is the rated speedThe circumferential force 119865119905 is related to the gear meshingstiffness detailed in ISO-6636-12006 [21] and the radial force119865119903 acting directly on the shaft which will affect the bearingdynamic coefficients determined primarily bearing enduredforce In a word we can conclude that the bearing dynamiccoefficients axial force and torque and gearmeshing stiffnesschange with the workload variation

311 Effect of Bearing Dynamic Coefficients In this paperthe unbalance response is linearly analyzed to determinethe resonance speed The existing program Dyrobes-BePerfhas been developed to analyze the dynamic performanceof tilting pad hydrodynamic journal bearings based on theFE Method The linear bearing stiffness coefficients anddamping coefficients can be calculated by this programFigure 9 shows the effect of increasing workload on thebearings of the output shafts As the workload increasesthe stiffness coefficients and damping coefficients of eachshaft bearing in 119909-direction and 119910-direction are increasingTo demonstrate the effect of bearing dynamic coefficientsunbalance responses of three output shafts only consideringthe increasing stiffness coefficients and damping coefficientsare presented in Figure 10 The critical speed of three outputshafts is markedly increased with the increase of the dynamiccoefficients of the bearings The low order critical speed ismainly affected from the results of the calculation

312 Effect of the Axial Force and Torque Figure 11 showsthe effect of increasing workload on the three output shaftsThe output shafts will not be subjected to axial force and

torque without the workload based on the above formula ofgear endured forceThe dynamic characteristics of the outputshafts are affectedwhen theworkload increases the axial forceand torque of the output shafts Figure 12 shows the unbalanceresponses of three output shafts considering the axial forceand torque of different workloads The critical speeds of theoutput shafts are slightly reduced as workload increases

313 Effect of the Gear Meshing Stiffness In ISO-6636-12006[21] the gear meshing stiffness is mainly affected by thespecific workload during the operation of the gear couplingsystem and the calculation formula of gear meshing stiffnessis given Based on the formula the gear meshing stiffnessof three output shafts under different workload is calculatedThe results are shown in Figure 13 The unbalance responsesof three output shafts are plotted (Figure 14) to investigatethe effect of gear meshing stiffness on output shafts Asthe gear meshing stiffness increases the lower-order criticalspeed increases slightly and the higher-order critical speedincreases obviously

The dynamic performance of the three output shaftscan be influenced by the bearing dynamic coefficients theaxial force and torque and the gear meshing stiffness Thecritical speeds of the output shafts significantly change withthe bearing dynamic coefficients and gear meshing stiffnessvaries These changes are slightly affected by axial force andtorque variation

32 Critical LoadAnalysis of the Geared Rotor-Bearing SystemFigure 15 shows the unbalance response of three output shaftsconsidering all the influencing factors under the varying

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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

8 Shock and Vibration

020

4060

80100

00511522520

4060

80

1152 Load ()Re

spon

se (120583

m)

100

n (rpm)times104

(a)

020

4060

80100

00511522520

4060

80

1152Load (

)Resp

onse

(120583m

)

100

n (rpm)times104

(b)

020

4060

80100

005115225Load (

)Resp

onse

(120583m

)

100

n (rpm)times104 20

4060

80

Load ()

115204

(c)

Figure 10 Unbalance response of three output shafts only considering dynamic coefficients under varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

0 20 40 60 80 100Load ()

T (N

m)

O1O2O3

104

103

102

(a)

O1O2O3

Fa

(N)

102

103

104

20 40 60 80 1000Load ()

(b)

Figure 11 Effect of increasing workload on the three output shafts (a) axial torque (b) axial force

workloadThe increase of the workload results in the increaseof the critical speed of output shafts The results illustratethat the rated speed below the first critical speed will befar away from the critical speed with the critical speed

increased Therefore the critical load unlikely occurs whenthe rated speed of each shaft is below the first critical speedConversely output shafts are generally running above thecritical speed in the integrally geared compressor and the

Shock and Vibration 9

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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

Load

()

Resp

onse

(120583m

)

100

n (rpm)152 1 05 025

times104

0

20

40

60

80

100

(a)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(b)

Resp

onse

(120583m

)

100

times104

Load

()

0

20

40

60

80

100

2 15 1 05 025n (rpm)

(c)

Figure 12 Unbalance response of three output shafts only considering axial torque and axial force under varying workload (a) output shaft1 (b) output shaft 2 (c) output shaft 3

K (N

m)

O1O1O2

20 40 60 80 1000Load ()

109

Figure 13 Effect of increasing workload on the gear meshing stiffness of three output shafts

10 Shock and Vibration

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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

020

4060

80100

005115225Load (

)

n (rpm)

2040

6080

1152Load (

)Resp

onse

(120583m

)

100

times104

(a)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(b)

020

4060

80100

005115225Load (

)

n (rpm)

Resp

onse

(120583m

)

100

times104

(c)

Figure 14 Unbalance response of three output shafts only considering gear meshing stiffness under varying workload (a) output shaft 1 (b)output shaft 2 and (c) output shaft 3

020

4060

80100

01

2Load ()

2040

6080

12

Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(a)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(b)

020

4060

80100

01

2Load ()

n (rpm)

Resp

onse

(120583m

)

100

times104 0515

25

(c)

Figure 15 Unbalance response of three output shafts considering all the influence factors under the varying workload (a) output shaft 1 (b)output shaft 2 (c) output shaft 3

critical speed can be adjusted to the rated speed of the outputshaft by increasing the workload Therefore the resonancemay occur under the influence of the workload when theoutput shafts are running at the rated speeds Figure 16illustrates the response amplitude and phase of the outputshafts under varying workload A slight resonance peak with

a slight phase change is observed in Figure 16(a) Whenthe workload is close to 75 the resonance peak and itsphase are obviously changed in Figure 16(b) Figure 16(c)displays that two obvious resonance peaks and their phaseschange significantly when the loads are close to 3 and 25respectively These outcomes demonstrate that critical load

Shock and Vibration 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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 11Ph

ase (

degr

ees)

Resp

onse

(m

) 101

20 40 60 80 1000Rotor load ()

minus200

0

200

20 40 60 80 1000Rotor load ()

(a)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

minus200

0

200

(b)

Phas

e (de

gree

s)Re

spon

se (

m)

100

101

minus200

0

200

20 40 60 80 1000Rotor load ()

20 40 60 80 1000Rotor load ()

(c)

Figure 16 Response and phase under different workloads when three output shafts are running at the rated speed (a) output shaft 1 runningat 11131 rmin (b) output shaft 2 running at 13297 rmin (c) output shaft 3 running at 20841 rmin

phenomenon occurs in O2 and O3 when they are at the ratedspeed

Our results reveal that the dynamic characteristics of thegeared rotor-bearing system change as workload varies Crit-ical load phenomenon likely occurs as workload is adjustedin the integrally geared compressor described in this paper

4 Conclusions

This study investigates the dynamic behaviors of the gearedrotor-bearing system in the integrally geared compressorinfluenced by varying workloads This study also examinesand discusses the effects of the bearing dynamic coefficientsaxial force and torque and gear meshing stiffness on thegeared rotor-bearing system under varying workloads Thecritical load phenomenon of the geared rotor-bearing systemis also analyzed in detail Results are summarized as follows

(1) The results demonstrate that the bearing dynamiccoefficients axial force and torque and gear meshing

stiffness are improved as workload increases Thebearing dynamic coefficients and gear meshing stiff-ness significantly affect the dynamic response of thesystem By contrast the influence of axial force andtorque is not significant

(2) Even though the rotors of the system do not have crit-ical speed close to the rated speed under no workloadconditions the geared rotor-bearing system producesresonance with the adjusted workload which is thecritical load phenomenon Similar to critical speedcritical load impedes the safe operation of the gearedrotor-bearing system

(3) The proposed method can be generally applied to theanalysis of the critical load which should be consid-ered in the designs of integrally geared compressorsThrough the analysis of the critical load we can adjustthe rated speed to avoid the critical load failure soas to improve the operational safety of the integrallygeared compressors

12 Shock and Vibration

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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

Appendix

A Finite Rotor Element Matrices

119872119890119879 = 120583119897119890420

sdot

[[[[[[[[[[[[[[[[[[

156 sym0 1560 minus22119897119890 4119897211989022119897 0 0 4119897211989054 0 0 13119897119890 1560 54 minus13119897119890 0 0 1560 13119897119890 minus31198972119890 0 0 22119897119890 41198972119890

minus13119897119890 0 0 minus31198972119890 minus22119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119872119890119877 = 1205831198901199031198902120119897119890

sdot

[[[[[[[[[[[[[[[[[[

36 sym0 360 minus3119897119890 411989721198903119897119890 0 0 41198972119890minus36 0 0 minus3119897119890 360 minus36119897119890 31198972119890 0 0 360 minus3119897119890 1198972119890 0 0 3119897119890 411989721198903119897119890 0 0 minus1198972119890 minus3119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]

119866119890 = 21205831198901199032119890120119897119890

sdot

[[[[[[[[[[[[[[[[[[[[

0 shew sym

36 0minus3119897119890 0 00 minus3119897119890 41198972119890 00 36 minus3119897119890 0 0minus36 0 0 minus3119897119890 36 0minus3119897119890 0 0 1198972119890 3119897119890 0 00 minus3119897119890 minus1198972119890 0 0 3119897119890 41198972119890 0

]]]]]]]]]]]]]]]]]]]]

119870119890119861 = 1198641198901198681198901198973119890

[[[[[[[[[[[[[[[[[[[

12 sym

0 120 minus6119897119890 411989721198906119897119890 0 0 41198972119890minus12 0 0 minus6119897119890 120 minus12 6119897119890 0 0 120 minus6119897119890 21198972119890 0 0 6119897119890 411989721198906119897119890 0 0 21198972119890 minus6119897119890 0 0 41198972119890

]]]]]]]]]]]]]]]]]]]

(A1)

B Gear Meshing Stiffness Matrix

119870119892 = 119896119892cos2 120573

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

sin2 120572 sin120572 cos120572 0 0 minussin2 120572 minus sin120572 cos120572 0 0 1199031 sin120572 1199032 sin120572sin120572 cos120572 cos2 120572 0 0 minus sin120572 cos120572 minuscos2 120572 0 0 1199031 cos120572 1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

minussin2 120572 minus sin120572 cos120572 0 0 sin2 120572 sin120572 cos120572 0 0 minus1199031 sin120572 minus1199032 sin120572minus sin120572 cos120572 minuscos2 120572 0 0 sin120572 cos120572 cos2 120572 0 0 minus1199031 cos120572 minus1199032 cos120572

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

1199031 sin120572 1199032 sin120572 0 0 minus1199031 sin120572 minus1199032 sin120572 0 0 11990321 119903111990321199031 cos120572 1199032 cos120572 0 0 minus1199031 cos120572 minus1199032 cos120572 0 0 11990311199032 11990322

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(B1)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge the financial supported by theNational Basic Research Program of China (Grant no

Shock and Vibration 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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 13

2012CB026000) and the National Science Foundationfor Distinguished Young Scholars of China (Grant no51305020)

References

[1] S Sutar and S Kadam ldquoAnalysis of geared rotorsmdasha reviewrdquoInternational Journal of Emerging Technology and AdvancedEngineering vol 2 no 10 pp 422ndash424 2012

[2] H D Nelson and J M McVaugh ldquoThe dynamics of rotor-bearing systems using finite elementsrdquo Journal of Engineeringfor Industry vol 98 no 2 pp 593ndash600 1976

[3] E S Zorzi and H D Nelson ldquoFinite element simulationof rotor-bearing systems with internal dampingrdquo Journal ofEngineering for Power vol 99 no 1 pp 71ndash76 1977

[4] E S Zorzi and H D Nelson ldquoDynamics of rotor-bearingsystems with axial torquemdasha finite element approachrdquo Journalof Mechanical Design vol 102 no 1 pp 158ndash161 1980

[5] A Kahraman H N Ozguven D R Houser and J J ZakrajsekldquoDynamic analysis of geared rotors by finite elementsrdquo Journalof Mechanical Design vol 114 no 3 pp 507ndash514 1992

[6] J S Rao T N Shiau and J R Chang ldquoTheoretical analysisof lateral response due to torsional excitation of geared rotorsrdquoMechanismandMachineTheory vol 33 no 6 pp 761ndash783 1998

[7] J-S Wu and C-H Chen ldquoTorsional vibration analysis of gear-branched systems by finite element methodrdquo Journal of Soundand Vibration vol 240 no 1 pp 159ndash182 2001

[8] S H Choi J Glienicke D C Han and K Urlichs ldquoDynamicgear loads due to coupled lateral torsional and axial vibrationsin a helical geared systemrdquo Journal of Vibration and Acousticsvol 121 no 2 pp 141ndash148 1999

[9] A S Lee J W Ha and D-H Choi ldquoCoupled lateral andtorsional vibration characteristics of a speed increasing gearedrotor-bearing systemrdquo Journal of Sound and Vibration vol 263no 4 pp 725ndash742 2003

[10] M Kubur A Kahraman D M Zini and K Kienzle ldquoDynamicanalysis of a multi-shaft helical gear transmission by finiteelements Model and experimentrdquo Journal of Vibration andAcoustics vol 126 no 3 pp 398ndash406 2004

[11] A S Lee and J W Ha ldquoPrediction of maximum unbalanceresponses of a gear-coupled two-shaft rotor-bearing systemrdquoJournal of Sound and Vibration vol 283 no 3ndash5 pp 507ndash5232005

[12] C H Kang W C Hsu E K Lee and T N Shiau ldquoDynamicanalysis of gear-rotor system with viscoelastic supports underresidual shaft bow effectrdquoMechanism and Machine Theory vol46 no 3 pp 264ndash275 2011

[13] Q Han J Zhao and F Chu ldquoDynamic analysis of a geared rotorsystem considering a slant crack on the shaftrdquo Journal of Soundand Vibration vol 331 no 26 pp 5803ndash5823 2012

[14] Y Zhang Q Wang H Ma J Huang and C Zhao ldquoDynamicanalysis of three-dimensional helical geared rotor system withgeometric eccentricityrdquo Journal of Mechanical Science and Tech-nology vol 27 no 11 pp 3231ndash3242 2013

[15] A Kahraman and R Singh ldquoInteractions between time-varyingmesh stiffness and clearance non-linearities in a geared systemrdquoJournal of Sound and Vibration vol 146 no 1 pp 135ndash156 1991

[16] A Al-Shyyab and A Kahraman ldquoNon-linear dynamic analysisof a multi-mesh gear train using multi-term harmonic balancemethod period-one motionsrdquo Journal of Sound and Vibrationvol 284 no 1-2 pp 151ndash172 2005

[17] C Siyu T Jinyuan L Caiwang and W Qibo ldquoNonlineardynamic characteristics of geared rotor bearing systems withdynamic backlash and frictionrdquo Mechanism and Machine The-ory vol 46 no 4 pp 466ndash478 2011

[18] J J Gao Diagnosis and Self-healing of Mechanical FailureHigher Education Press Beijing China 2012

[19] A Vollan and L Komzsik Computational Techniques of RotorDynamics with the Finite Element Method CRC Press NewYork NY USA 2012

[20] Standard ISO 1940-1 2003 Mechanical VibrationmdashBalanceQuality Requirements for Rotors in aConstant (rigid) StatemdashPart1 Specification and Verification of Balance Tolerances Interna-tional Organization for Standardization Geneva Switzerland2003

[21] ISO ldquoCalculation of load capacity of spur and helical gearsmdashpart 1 basic principles introduction and general influence fac-torsrdquo ISO Standard 6336-12006 International Organizationfor Standardization Geneva Switzerland 2006

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 athttpwwwhindawicom

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 athttpwwwhindawicom

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