Research ArticleEffects of Surface Waviness on the Nonlinear Vibration of GasLubricated Bearing-Rotor System

Jian Li12 Shaoqi Yang13 Xiaoming Li13 and Qing Li 123

1State Key Laboratory of Technologies in Space Cryogenic Propellants Beijing 100190 China2School of Energy Power and Engineering Huazhong University of Science and Technology Wuhan Hubei 430074 China3University of Chinese Academy of Sciences Beijing 100049 China

Correspondence should be addressed to Qing Li simple_lxcy2014163com

Received 30 May 2018 Accepted 25 September 2018 Published 4 November 2018

Academic Editor Konstantin Avramov

Copyright copy 2018 Jian Li et al -is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-is paper presents the effects of the surface waviness on the nonlinear dynamic performance of a gas bearing-rotor system -ecoupled vibration with the elastomer is taken into consideration to fit the actual engineering application -e effects of thedirections the amplitudes and the numbers of waves to the nonlinear dynamic performance are investigated -e results showthat the existence of the surface waviness in the circumferential direction can improve the stability of the system obviously But thesurface waviness in the axial direction is damage to the system-e nonlinear dynamic performance of the system is insensitive tothe number of surface waviness -e increase of the amplitude of the waviness in circumferential direction can improve thestability of the system

1 Introduction

Bearing is an essential component of most rotatingmachines -ey provide the load capacity and stiffness tosupport shafts against static loads and dynamic forcesGas lubricating journal bearings have many advantagesof less heat generation more environment friendlyhigher accuracy and longer lives by comparing withtraditional oil bearings -us the gas lubricatingbearing-rotor system is widely used in engineering ap-plications such as turboexpanders in large-scale cryo-genic systems superprecision machine tools and high-speed machines [1 2] However the inherent charac-teristic of compressibility and low viscidity of gas maycause instability to the gas bearing-rotor system whichlimits the range of the application of gas bearingsFurthermore the low viscosity of gas makes a thin gasfilm generally lower than 50 micrometers As the resultthe assumption of the journal bearing is that the smoothsurface is far from bearing practical application espe-cially for bearing-rotor system operating under a rela-tively small film thickness [3]

Manufacturing error may cause some waves on thesurfaces of bearing -ese waves can change the thicknessdistribution of the gas film which can significantly influencethe static and dynamic performance of the gas bearing-rotorsystem During the past years authors have published manyarticles about the effects of the surface waviness on theperformance of the bearing Dimofte [4] compared thethree-waved journal bearing with both the three-wave-groove bearing and three-lobe bearing [5] they concludedthat the load capacity of the wave bearing is better than othertypes and the performance of the wave bearing is dependenton the parameters and position of waves Rasheed [3]studied the influence of circumferential axial and combinedsurface waviness on the performance of plain cylindricalbearings -ey noted that the circumferential waviness canincrease the load capacity and decrease the friction variablewhen the wave number is below 9 and the axial waviness hasthe adverse effects on load capacity and friction Kwan andPost [6] studied the influence of manufacturing error ofrectangular thrust bearing on the static load and stiffnessperformance -ey concluded that the load capacity andstiffness are sensitive to the manufacturing errors Nagaraju

HindawiShock and VibrationVolume 2018 Article ID 8269384 16 pageshttpsdoiorg10115520188269384

et al [7] studied the combined influence of bearing flexibilityand surface roughness on the characteristics of the journalbearing system -e results indicated that the longitudinalroughness pattern of the bearing can improve the loadcapacity Ene et al [8] used both the transient and smallperturbationmethods to investigate the dynamic behavior ofthree-wave journal bearing system -e results showed theinfluence of the wave amplitude and oil supply pressure onthe threshold of stability Sharma and Kushare [9] in-vestigated the influence of surface roughness on the per-formance of two-lobe journal bearing -e results indicatedthat the performance of the bearing system is significantlyaffected by the orientation of roughness And thus theperformance of bearing can be enhanced by the properselection of the roughness pattern parameter Wang et al[10] studied the influence of the waves of bearing surface onthe static and dynamic performance of the aerostatic journalbearing -e results showed that the wave amplitudes sig-nificantly affect the static and dynamic performance of thebearing

-e stability is the most important character for thebearing-rotor system Large number of articles has beenpublished to investigate the stability of gas-lubricantbearing-rotor system Among these papers two differentapproaches were published to analyze the dynamic stabilityof the rotor-bearing system one is the linear small per-turbation analysis and another is the nonlinear transientanalysis

Linear small perturbation theory assumes that thejournal obit whirls around its equilibrium position witha small perturbation -is assumption makes the dynamicstiffness and dynamic damping coefficients can be com-puted And then the whirl frequency and critical mass canbe computed by these dynamic coefficients-is method canprovide the instability boundary -us the stability of thesystem can be estimated by the threshold However thismethod cannot present the characters of the motion

Gross and Zachmanoglou [11] studied the performanceof the self-acting journal bearing with infinite length byusing the small perturbation method -e calculation resultsare valid for other types of bearings with high accuracyBelforte et al [12] investigated dynamic coefficients of self-acting gas bearing they used the developed mathematicalmodel and the results were in good agreement with theresults using the nonlinear distribution parameter modelYang et al [13] studied the dynamic coefficients of self-acting tilting pad gas bearing and the results showed therelationship between dynamic coefficients and tilting padperturbation Yu et al [14] used numerical simulations andexperiments to investigate the influence of perturbations onthe dynamic stiffness for aerostatic bearings -e resultsrevealed that the dynamic stiffness coefficients are moresensitive to the frequency of perturbation than the ampli-tude thus they concluded that the dynamic stiffness can beenhanced through active control which can generateperturbations

In nonlinear transient analysis the time-dependentReynolds equation is solved to get the transient forcesand then the transient journal center obits are obtained by

solving the motion equations -e stability of the system canbe judged by the shape of obits -e nonlinear dynamiccharacters are available by using this method which is dif-ferent from the perturbation analysis

Kim and Noah [15] coupled harmonic balance withalternating frequency technique to obtain synchronous andsubsynchronous whirling motions of a Jeffcott rotor-bearingsystem -e results showed that as the parameters of thesystem changed the boundary of flip bifurcation were ob-tained Wang et al [16ndash20] investigated the bifurcationperformance of rigid rotor and flexible rotor supported bya pair of gas bearing and the results presented a betterunderstanding to the nonlinear dynamic characters of thegas bearing-rotor system From the calculation results wecan also find that the dynamic behavior of the rotor center isvery complex which includes periodic and subharmonicresponse Rashidi et al [1 2] presented the effect of operatingparameters on nonlinear dynamic performance for non-circular journal gas-lubricated bearing-rotor system -ecomplex nonlinear dynamic behaviors such as periodic KT-periodic and quasiperiodic responses were analyzed byusing dynamic trajectory the Poincare maps and bi-furcation diagrams Hassini and Arghir [21] used the dy-namic coefficients to analyze the nonlinear characteristicsand the results were in good agreement with the resultscalculated by nonlinear transient Reynolds equationsAbbasi et al [22] studied the vibration control of the rotatingmachine with a high-static low-dynamic stiffness suspen-sion In their study the complex nonlinear dynamic be-havior was showed and the results present that the methodcan suppress the nonperiodic behavior in the consideredspeed range

Generally speaking the vibrations of most manufacturemachines are inevitable which will cause the surface to beunsmooth -e effects of waves on static performances anddynamic coefficients of bearing are investigated in mostprevious published articles However few papers havementioned the effects of waves on the stability and nonlineardynamic performance of the gas bearing-us in this paperthe effects of surface waviness in circumferential and axialdirections on the nonlinear dynamic characteristics of therotor-gas bearing system have been investigated -emethod used in this paper is nonlinear transient approachwhich is similar to the previous articles [18 23] but ourfocuses are different In practical engineering the elastomersare mounted between bearing and shaft thus the elastomersuspension is taken into consideration -e results in thispaper can provide guidance on the design and manufactureof the gas bearing system

2 Numerical Solution of GasPressure Distribution

-e model in this paper is a massless shaft with a disksupported by a pair of self-acting gas journal bearingsGenerally speaking the manufacture precision of rotors isalways better than that of bearings so the waves on thesurface of rotors can be neglected [22] In this paper thejournal bearings used for calculation include axial and

2 Shock and Vibration

circumferential surface waviness which is shown in Figure 1Before numerical simulation some design assumptions aregiven as follows [16ndash18 23]

(1) Isothermal flow due to the heat conductivity abilityof the bearing material (copper) is much bigger thanthat of gas the heat generation in the gas film willconduct away quickly-us the temperature of gas isconstant

(2) Curvature insensitive the gas film thickness is muchsmaller than the diameter of the shaft So the changeof velocity caused by the curvature is neglected

(3) Constant viscosity due to the viscosity of gas isinsensitive to the changes of pressure we assume it asa virtually constant

(4) -e side flow of the bearing can be neglected

To simplify the model of surface waves we assume thatthe form of waves fits the function of sinusoidal thus thewaviness in circumferential direction can be described as thefollowing form [10 24]

δ A sin2πR

λθ1113874 1113875 (1)

-e waves in axial direction of the bearing can bedenoted as

δ A sin2πλ

y1113874 1113875 (2)

Based on the assumption above the pressure distribu-tion of the gas film can be described by the Reynoldsequation

z

zx

ph3

μzp

zx1113888 1113889 +

z

zy

ph3

μzp

zy1113888 1113889 6 U

z

zx(ph) + 2

z

zt(ph)1113890 1113891

(3)

where h is the gas film thickness and can be denoted as

h Cr(1 + ε cosφ)minus δ (4)

Some dimensionless parameters are defined as follows totransform Equation (3) into polar coordinate andnondimensionalized

P p

Pa

U Rϖ

τ ϖt

Λ 6μϖPa

R

Cr1113874 1113875

2

H h

Cr

θ x

R

η y

R

(5)

Taking the dimensionless parameters above into Equa-tion (3) the dimensionless Reynolds equation is obtainedand can be written as

z

zθPH

3zP

zθ1113888 1113889 +

z

zηPH

3zP

zη1113888 1113889 Λ

z

zθ(ph) + 2Λ

z

zτ(ph)

(6)

-e finite difference method is used to get the pressuredistribution of the gas film -e discretized scheme of theReynolds equation was derived by using the central differ-ence method in circumferential and axial directions and theimplicit backwards method in time τ -e methods used inthis paper are similar to the previous articles [18 23] and thediscretized version can be described as

3 Hn+1i( 1113857

2

2Hn+1

i+1 minusHn+1iminus1( 1113857

2ΔθSn+1

i+1j minus Sn+1iminus1j1113872 1113873

2Δθ+ H

n+1ij1113872 1113873

3 Sn+1i+1j minus 2Sn+1

ij + Sn+1iminus1j

(Δθ)2⎛⎝ ⎞⎠ +

Hn+1i( 1113857

3

2Sn+1

ij+1 minus 2Sn+1ij + Sn+1

ijminus11113872 1113873

(Δη)2

ΛPn+1ij

Hn+1i+1 minusHn+1

iminus12Δθ

1113888 1113889 + ΛHn+1i

Pn+1i+1j minusPn+1

iminus1j

2Δθ1113888 1113889 + 2ΛHn+1

i

Pn+1ij minusPn

ij

Δτ1113888 1113889 + 2ΛPn+1

ij

Hn+1ij minusHn

ij

2Δτ1113888 1113889

(7)

where S P2 i 1 2 N j 1 2 M-e film thickness distribution is depended by the

position of the rotor For parameter Pn+1ij the equation is

a quadratic equation with one unknown and determinedby the pressure of the surrounding four nodes (iminus 1 j)(i + 1 j) (i j + 1) and (i jminus 1)

-e pressure distribution of the gas film between therotor and bearing enables the estimation of the supporting

force which is acting on the rotor -e supporting force canbe described as

fgx PaR2

1113946LR

minus(LR)11139462π

0P(θ η) cos θ dθ dη

fgy PaR2

1113946LR

minus(LR)11139462π

0P(θ η) sin θ dθ dη

(8)

Shock and Vibration 3

where P(θ η) is the pressure distribution in the di-mensionless coordinate system

3 Rotor Dynamic Equations and ObitsCalculation Procedure

In this paper we use the nonlinear transient approach toinvestigate the stability of the rotor-bearing system -emodel for calculation is a massless rotor with a disk and itwas supported by a pair of self-acting gas bearings Inengineering reality the elastic dampers such as O-rings aremounted between bearings and shell to improve its sta-bility shown in Figure 2 -e damping force of the elas-tomer obeys the viscous damping law which is one of thesimplest and most widely used -is model is expressed as[25ndash27]

fd sign _xb( 1113857c _xb1113868111386811138681113868

1113868111386811138681113868c (9)

where c is the damping exponent sign() is the signumfunction -e case c 1 is chose which indicates the linearviscous damping model and used for calculation

-us the equation of motion of the rotor and bearing intransient state can be written as

msd2xs

dt2 fex minusfgx + mseubw

2 sin(wt)

msd2yr

dt2 fey minusfgy + mseubw

2 cos(wt)

mbd2xb

dt2 fgx minus kxb minus sign _xb( 1113857c _xb

11138681113868111386811138681113868111386811138681113868c

mbd2yb

dt2 fgy minus kyb minus sign _yb( 1113857c _yb

11138681113868111386811138681113868111386811138681113868c

(10)

-e nondimensional groups can be defined as follows

Xs xs

Cr

Ys ys

Cr

Xb xb

Cr

Yb yb

Cr

Fex fex

PaRL

Fey fey

PaRL

Fgx fgx

PaRL

Fgy fgy

PaRL

Ms msCrw2

PaRL

Mb mbCrw2

PaRL

ρ eub

Cr

K kCr

PaRL

C c(Crw)c

PaRL

(11)

Substituting Equation (11) into Equation (10) we will getthe dimensionless dynamic equations as follows

Bearing

RotorL

e

d D

ω

Figure 1 Model of self-acting gas journal bearing with surface waviness

4 Shock and Vibration

d2Xs

dτ2

Fex minusFgx + Msρ sin(τ)

Ms Asx

d2Ys

dτ2

Yex minusYgx + Msρ cos(τ)

Ms Asy

d2Xb

dτ2

Fgx minusKXb minus sign( _X)C _Xb1113868111386811138681113868

1113868111386811138681113868c

Mb Abx

d2Yb

dτ2

Ygx minusKYb minus sign( _Y)C _Yb1113868111386811138681113868

1113868111386811138681113868c

Mb Aby

(12)

As we get acceleration of the rotor and bearing inhorizontal and vertical directions the center obits of rotorand bearing can be acceptable by using the followingequation

Vsx Vsx0 + AsxΔτ

Vsy Vsy0 + AsyΔτ

Vbx Vbx0 + AbxΔτ

Vby Vby0 + AbyΔτ

Xs Xs0 + VsxΔτ +12AsxΔτ

2 Ys

Ys0 + VsyΔτ +12AsyΔτ

2 Xb

Xb0 + VbxΔτ +12AbxΔτ

2 Yb Yb0 + VbyΔτ +

12AbyΔτ

2

(13)

Computation of the center trajectory of rotor andbearing is an iterative procedure -e procedure begins withthe initial equilibrium state -e initial position of the rotor(X0 Y0) is a static balance position which is defined by theexternal loading and rotational speed To estimate the in-fluence of the initial condition the first 600 periodic cal-culation data can be ignored and the nonlinear behavioranalysis is based on the data after the vibration is stable -einitial velocities of the rotor in two different directions areassumed to be zero

4 Results and Discussion

In order to verify the correctness of the program the attitudeangle φ and supporting force F of a self-acting journal gasbearing with smooth surface are calculated and compared

Rotor Journal bearing

Elastomer suspension

ωOs

Ob Dde

(a)

Rotor

Elastomer suspension

Journal bearing

Disk

(b)

Figure 2 Schematic of a rigid rotor (massless shaft and disk) in journal bearing with elastic damper

Shock and Vibration 5

with previous investigations [23 28] as is shown in Figure 3-e results in this paper are in good agreement with theprevious investigations thus the validity of the program isverified

-e rotor unbalance mass is an important parameter forrotating machine which influence the response of the rotor-bearing system [28 29] In this paper what we focused on isthe influence of the surface waviness to the nonlinear dy-namic response So the single rotor unbalance value(015 μm) is selected in this study -e rotor-bearing systemis used in the turbomachine with a superhigh speed -usthe mass of the rotor is usually quite small less than 05 kg

In the numerical integration the aspect ratio (LD) ofself-acting journal gas bearing for calculation is chosen as144 -e lubricant in the film is air -e parameters ofelastomer are set as follows k 4e6Nm ξ 0005 and c 1To show the influence of the surface waviness to the non-linear dynamic response clearly the bifurcation diagrams ofthe rotor-bearing system with flat surface are plotted whichis presented in Figure 4 From the results we can concludethat when the half mass of the rotor is equal or less than018 kg the rotor center presents a regular periodic motionBut motion loses its regularity and becomes irregular whenthe half mass of the rotor increased to 019 kg

Figure 5 shows the bifurcation diagrams of the rotatingsystem with circumferential surface waviness In this cal-culation we assume that the surface waviness is only in-cluded in the circumferential direction of the bearing andthe number of waves is equal to 5 and the amplitude of thewaves is 1 micrometer Compared with the results in Fig-ure 4 the discrete point of the system with circumferentialwaviness is much bigger than the system with flat surfacewhich indicated that the waviness in circumferential di-rection can increase the stability of the system obviously

In order to show nonlinear dynamic performance of thesystem clearly a set of maps such as the rotor center obits

phase portraits logarithmic spectra maps in two directionsand Poincare maps are used to analyze the nonlinear dy-namic performance in different conditions

Figures 6(a)ndash13(a) are the obits of the rotor center in the (XY) plane Figures 6(b)ndash13(b) and 6(c)ndash13(c) are phase portraitsin (X _X) and (Y _Y) planes respectively Figures 6(d)ndash13(d)and 6(e)ndash13(e) are the logarithmic spectra of the non-dimensional displacement in horizontal and vertical re-spectively Figures 6(f)ndash13(f) are Poincare maps in the (X Y)plane

From Figures 6ndash8 when the half mass of the rotor is lessthan 075 kg the vibration is mainly caused by the massunbalance the frequency of rotating is much bigger thansubharmonic frequency thus the rotor center obits showa regular periodic motion When the half mass of the rotor isin the range of 075 kg and 1 kg the amplitude of sub-harmonic vibration increased and cannot be ignored -emotion losses its periodic motion and subharmonic peri-odic motion appears However the amplitude of motion isalso small When the half mass of the rotor is bigger than10 kg shown in Figures 8 and 9 the amplitude of vibrationbecomes much bigger and a closed curve occurs in thePoincare map which indicated that quasiperiodic motionappears

-e effects of the number of waviness in the circum-ferential direction on the nonlinear dynamic performance ofthe system are analyzed in which different numbers ofwaviness (N 3 5 and 7) with ms 10 kg ξ 0005 k

4e6Nm are used for calculation Figures 8ndash11 show theobits logarithmic spectrum maps and Poincare map withsurface waviness (N 3 5 and 7) respectively -e com-parison between different numbers of waviness in circum-ferential direction shows that the number of waviness do notchange the nonlinear dynamic performance of the systembut the amplitude of the vibration declines slightly with theincrease of the number of waviness (shown in the

01 02 03 04 05 06 07 08Eccentricity ratio

01 02 03 04 05 06 07 08Eccentricity ratio

Dim

ensio

nles

s for

ce

0

1

2

3

4

5

6

7

Atti

tude

angl

e(deg)

10

20

30

40

50

60

70

80

Λ = 06

Λ = 06

Λ = 30

Λ = 30

Reference [23]Reference [28]This paper

Reference [23]Reference [28]This paper

Figure 3 Comparison of steady state calculation results for journal gas bearing D L

6 Shock and Vibration

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

circumferential surface waviness which is shown in Figure 1Before numerical simulation some design assumptions aregiven as follows [16ndash18 23]

(1) Isothermal flow due to the heat conductivity abilityof the bearing material (copper) is much bigger thanthat of gas the heat generation in the gas film willconduct away quickly-us the temperature of gas isconstant

(2) Curvature insensitive the gas film thickness is muchsmaller than the diameter of the shaft So the changeof velocity caused by the curvature is neglected

(3) Constant viscosity due to the viscosity of gas isinsensitive to the changes of pressure we assume it asa virtually constant

(4) -e side flow of the bearing can be neglected

To simplify the model of surface waves we assume thatthe form of waves fits the function of sinusoidal thus thewaviness in circumferential direction can be described as thefollowing form [10 24]

δ A sin2πR

λθ1113874 1113875 (1)

-e waves in axial direction of the bearing can bedenoted as

δ A sin2πλ

y1113874 1113875 (2)

Based on the assumption above the pressure distribu-tion of the gas film can be described by the Reynoldsequation

z

zx

ph3

μzp

zx1113888 1113889 +

z

zy

ph3

μzp

zy1113888 1113889 6 U

z

zx(ph) + 2

z

zt(ph)1113890 1113891

(3)

where h is the gas film thickness and can be denoted as

h Cr(1 + ε cosφ)minus δ (4)

Some dimensionless parameters are defined as follows totransform Equation (3) into polar coordinate andnondimensionalized

P p

Pa

U Rϖ

τ ϖt

Λ 6μϖPa

R

Cr1113874 1113875

2

H h

Cr

θ x

R

η y

R

(5)

Taking the dimensionless parameters above into Equa-tion (3) the dimensionless Reynolds equation is obtainedand can be written as

z

zθPH

3zP

zθ1113888 1113889 +

z

zηPH

3zP

zη1113888 1113889 Λ

z

zθ(ph) + 2Λ

z

zτ(ph)

(6)

-e finite difference method is used to get the pressuredistribution of the gas film -e discretized scheme of theReynolds equation was derived by using the central differ-ence method in circumferential and axial directions and theimplicit backwards method in time τ -e methods used inthis paper are similar to the previous articles [18 23] and thediscretized version can be described as

3 Hn+1i( 1113857

2

2Hn+1

i+1 minusHn+1iminus1( 1113857

2ΔθSn+1

i+1j minus Sn+1iminus1j1113872 1113873

2Δθ+ H

n+1ij1113872 1113873

3 Sn+1i+1j minus 2Sn+1

ij + Sn+1iminus1j

(Δθ)2⎛⎝ ⎞⎠ +

Hn+1i( 1113857

3

2Sn+1

ij+1 minus 2Sn+1ij + Sn+1

ijminus11113872 1113873

(Δη)2

ΛPn+1ij

Hn+1i+1 minusHn+1

iminus12Δθ

1113888 1113889 + ΛHn+1i

Pn+1i+1j minusPn+1

iminus1j

2Δθ1113888 1113889 + 2ΛHn+1

i

Pn+1ij minusPn

ij

Δτ1113888 1113889 + 2ΛPn+1

ij

Hn+1ij minusHn

ij

2Δτ1113888 1113889

(7)

where S P2 i 1 2 N j 1 2 M-e film thickness distribution is depended by the

position of the rotor For parameter Pn+1ij the equation is

a quadratic equation with one unknown and determinedby the pressure of the surrounding four nodes (iminus 1 j)(i + 1 j) (i j + 1) and (i jminus 1)

-e pressure distribution of the gas film between therotor and bearing enables the estimation of the supporting

force which is acting on the rotor -e supporting force canbe described as

fgx PaR2

1113946LR

minus(LR)11139462π

0P(θ η) cos θ dθ dη

fgy PaR2

1113946LR

minus(LR)11139462π

0P(θ η) sin θ dθ dη

(8)

Shock and Vibration 3

where P(θ η) is the pressure distribution in the di-mensionless coordinate system

3 Rotor Dynamic Equations and ObitsCalculation Procedure

In this paper we use the nonlinear transient approach toinvestigate the stability of the rotor-bearing system -emodel for calculation is a massless rotor with a disk and itwas supported by a pair of self-acting gas bearings Inengineering reality the elastic dampers such as O-rings aremounted between bearings and shell to improve its sta-bility shown in Figure 2 -e damping force of the elas-tomer obeys the viscous damping law which is one of thesimplest and most widely used -is model is expressed as[25ndash27]

fd sign _xb( 1113857c _xb1113868111386811138681113868

1113868111386811138681113868c (9)

where c is the damping exponent sign() is the signumfunction -e case c 1 is chose which indicates the linearviscous damping model and used for calculation

-us the equation of motion of the rotor and bearing intransient state can be written as

msd2xs

dt2 fex minusfgx + mseubw

2 sin(wt)

msd2yr

dt2 fey minusfgy + mseubw

2 cos(wt)

mbd2xb

dt2 fgx minus kxb minus sign _xb( 1113857c _xb

11138681113868111386811138681113868111386811138681113868c

mbd2yb

dt2 fgy minus kyb minus sign _yb( 1113857c _yb

11138681113868111386811138681113868111386811138681113868c

(10)

-e nondimensional groups can be defined as follows

Xs xs

Cr

Ys ys

Cr

Xb xb

Cr

Yb yb

Cr

Fex fex

PaRL

Fey fey

PaRL

Fgx fgx

PaRL

Fgy fgy

PaRL

Ms msCrw2

PaRL

Mb mbCrw2

PaRL

ρ eub

Cr

K kCr

PaRL

C c(Crw)c

PaRL

(11)

Substituting Equation (11) into Equation (10) we will getthe dimensionless dynamic equations as follows

Bearing

RotorL

e

d D

ω

Figure 1 Model of self-acting gas journal bearing with surface waviness

4 Shock and Vibration

d2Xs

dτ2

Fex minusFgx + Msρ sin(τ)

Ms Asx

d2Ys

dτ2

Yex minusYgx + Msρ cos(τ)

Ms Asy

d2Xb

dτ2

Fgx minusKXb minus sign( _X)C _Xb1113868111386811138681113868

1113868111386811138681113868c

Mb Abx

d2Yb

dτ2

Ygx minusKYb minus sign( _Y)C _Yb1113868111386811138681113868

1113868111386811138681113868c

Mb Aby

(12)

As we get acceleration of the rotor and bearing inhorizontal and vertical directions the center obits of rotorand bearing can be acceptable by using the followingequation

Vsx Vsx0 + AsxΔτ

Vsy Vsy0 + AsyΔτ

Vbx Vbx0 + AbxΔτ

Vby Vby0 + AbyΔτ

Xs Xs0 + VsxΔτ +12AsxΔτ

2 Ys

Ys0 + VsyΔτ +12AsyΔτ

2 Xb

Xb0 + VbxΔτ +12AbxΔτ

2 Yb Yb0 + VbyΔτ +

12AbyΔτ

2

(13)

Computation of the center trajectory of rotor andbearing is an iterative procedure -e procedure begins withthe initial equilibrium state -e initial position of the rotor(X0 Y0) is a static balance position which is defined by theexternal loading and rotational speed To estimate the in-fluence of the initial condition the first 600 periodic cal-culation data can be ignored and the nonlinear behavioranalysis is based on the data after the vibration is stable -einitial velocities of the rotor in two different directions areassumed to be zero

4 Results and Discussion

In order to verify the correctness of the program the attitudeangle φ and supporting force F of a self-acting journal gasbearing with smooth surface are calculated and compared

Rotor Journal bearing

Elastomer suspension

ωOs

Ob Dde

(a)

Rotor

Elastomer suspension

Journal bearing

Disk

(b)

Figure 2 Schematic of a rigid rotor (massless shaft and disk) in journal bearing with elastic damper

Shock and Vibration 5

with previous investigations [23 28] as is shown in Figure 3-e results in this paper are in good agreement with theprevious investigations thus the validity of the program isverified

-e rotor unbalance mass is an important parameter forrotating machine which influence the response of the rotor-bearing system [28 29] In this paper what we focused on isthe influence of the surface waviness to the nonlinear dy-namic response So the single rotor unbalance value(015 μm) is selected in this study -e rotor-bearing systemis used in the turbomachine with a superhigh speed -usthe mass of the rotor is usually quite small less than 05 kg

In the numerical integration the aspect ratio (LD) ofself-acting journal gas bearing for calculation is chosen as144 -e lubricant in the film is air -e parameters ofelastomer are set as follows k 4e6Nm ξ 0005 and c 1To show the influence of the surface waviness to the non-linear dynamic response clearly the bifurcation diagrams ofthe rotor-bearing system with flat surface are plotted whichis presented in Figure 4 From the results we can concludethat when the half mass of the rotor is equal or less than018 kg the rotor center presents a regular periodic motionBut motion loses its regularity and becomes irregular whenthe half mass of the rotor increased to 019 kg

Figure 5 shows the bifurcation diagrams of the rotatingsystem with circumferential surface waviness In this cal-culation we assume that the surface waviness is only in-cluded in the circumferential direction of the bearing andthe number of waves is equal to 5 and the amplitude of thewaves is 1 micrometer Compared with the results in Fig-ure 4 the discrete point of the system with circumferentialwaviness is much bigger than the system with flat surfacewhich indicated that the waviness in circumferential di-rection can increase the stability of the system obviously

In order to show nonlinear dynamic performance of thesystem clearly a set of maps such as the rotor center obits

phase portraits logarithmic spectra maps in two directionsand Poincare maps are used to analyze the nonlinear dy-namic performance in different conditions

Figures 6(a)ndash13(a) are the obits of the rotor center in the (XY) plane Figures 6(b)ndash13(b) and 6(c)ndash13(c) are phase portraitsin (X _X) and (Y _Y) planes respectively Figures 6(d)ndash13(d)and 6(e)ndash13(e) are the logarithmic spectra of the non-dimensional displacement in horizontal and vertical re-spectively Figures 6(f)ndash13(f) are Poincare maps in the (X Y)plane

From Figures 6ndash8 when the half mass of the rotor is lessthan 075 kg the vibration is mainly caused by the massunbalance the frequency of rotating is much bigger thansubharmonic frequency thus the rotor center obits showa regular periodic motion When the half mass of the rotor isin the range of 075 kg and 1 kg the amplitude of sub-harmonic vibration increased and cannot be ignored -emotion losses its periodic motion and subharmonic peri-odic motion appears However the amplitude of motion isalso small When the half mass of the rotor is bigger than10 kg shown in Figures 8 and 9 the amplitude of vibrationbecomes much bigger and a closed curve occurs in thePoincare map which indicated that quasiperiodic motionappears

-e effects of the number of waviness in the circum-ferential direction on the nonlinear dynamic performance ofthe system are analyzed in which different numbers ofwaviness (N 3 5 and 7) with ms 10 kg ξ 0005 k

4e6Nm are used for calculation Figures 8ndash11 show theobits logarithmic spectrum maps and Poincare map withsurface waviness (N 3 5 and 7) respectively -e com-parison between different numbers of waviness in circum-ferential direction shows that the number of waviness do notchange the nonlinear dynamic performance of the systembut the amplitude of the vibration declines slightly with theincrease of the number of waviness (shown in the

01 02 03 04 05 06 07 08Eccentricity ratio

01 02 03 04 05 06 07 08Eccentricity ratio

Dim

ensio

nles

s for

ce

0

1

2

3

4

5

6

7

Atti

tude

angl

e(deg)

10

20

30

40

50

60

70

80

Λ = 06

Λ = 06

Λ = 30

Λ = 30

Reference [23]Reference [28]This paper

Reference [23]Reference [28]This paper

Figure 3 Comparison of steady state calculation results for journal gas bearing D L

6 Shock and Vibration

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

where P(θ η) is the pressure distribution in the di-mensionless coordinate system

3 Rotor Dynamic Equations and ObitsCalculation Procedure

In this paper we use the nonlinear transient approach toinvestigate the stability of the rotor-bearing system -emodel for calculation is a massless rotor with a disk and itwas supported by a pair of self-acting gas bearings Inengineering reality the elastic dampers such as O-rings aremounted between bearings and shell to improve its sta-bility shown in Figure 2 -e damping force of the elas-tomer obeys the viscous damping law which is one of thesimplest and most widely used -is model is expressed as[25ndash27]

fd sign _xb( 1113857c _xb1113868111386811138681113868

1113868111386811138681113868c (9)

where c is the damping exponent sign() is the signumfunction -e case c 1 is chose which indicates the linearviscous damping model and used for calculation

-us the equation of motion of the rotor and bearing intransient state can be written as

msd2xs

dt2 fex minusfgx + mseubw

2 sin(wt)

msd2yr

dt2 fey minusfgy + mseubw

2 cos(wt)

mbd2xb

dt2 fgx minus kxb minus sign _xb( 1113857c _xb

11138681113868111386811138681113868111386811138681113868c

mbd2yb

dt2 fgy minus kyb minus sign _yb( 1113857c _yb

11138681113868111386811138681113868111386811138681113868c

(10)

-e nondimensional groups can be defined as follows

Xs xs

Cr

Ys ys

Cr

Xb xb

Cr

Yb yb

Cr

Fex fex

PaRL

Fey fey

PaRL

Fgx fgx

PaRL

Fgy fgy

PaRL

Ms msCrw2

PaRL

Mb mbCrw2

PaRL

ρ eub

Cr

K kCr

PaRL

C c(Crw)c

PaRL

(11)

Substituting Equation (11) into Equation (10) we will getthe dimensionless dynamic equations as follows

Bearing

RotorL

e

d D

ω

Figure 1 Model of self-acting gas journal bearing with surface waviness

4 Shock and Vibration

d2Xs

dτ2

Fex minusFgx + Msρ sin(τ)

Ms Asx

d2Ys

dτ2

Yex minusYgx + Msρ cos(τ)

Ms Asy

d2Xb

dτ2

Fgx minusKXb minus sign( _X)C _Xb1113868111386811138681113868

1113868111386811138681113868c

Mb Abx

d2Yb

dτ2

Ygx minusKYb minus sign( _Y)C _Yb1113868111386811138681113868

1113868111386811138681113868c

Mb Aby

(12)

As we get acceleration of the rotor and bearing inhorizontal and vertical directions the center obits of rotorand bearing can be acceptable by using the followingequation

Vsx Vsx0 + AsxΔτ

Vsy Vsy0 + AsyΔτ

Vbx Vbx0 + AbxΔτ

Vby Vby0 + AbyΔτ

Xs Xs0 + VsxΔτ +12AsxΔτ

2 Ys

Ys0 + VsyΔτ +12AsyΔτ

2 Xb

Xb0 + VbxΔτ +12AbxΔτ

2 Yb Yb0 + VbyΔτ +

12AbyΔτ

2

(13)

Computation of the center trajectory of rotor andbearing is an iterative procedure -e procedure begins withthe initial equilibrium state -e initial position of the rotor(X0 Y0) is a static balance position which is defined by theexternal loading and rotational speed To estimate the in-fluence of the initial condition the first 600 periodic cal-culation data can be ignored and the nonlinear behavioranalysis is based on the data after the vibration is stable -einitial velocities of the rotor in two different directions areassumed to be zero

4 Results and Discussion

In order to verify the correctness of the program the attitudeangle φ and supporting force F of a self-acting journal gasbearing with smooth surface are calculated and compared

Rotor Journal bearing

Elastomer suspension

ωOs

Ob Dde

(a)

Rotor

Elastomer suspension

Journal bearing

Disk

(b)

Figure 2 Schematic of a rigid rotor (massless shaft and disk) in journal bearing with elastic damper

Shock and Vibration 5

with previous investigations [23 28] as is shown in Figure 3-e results in this paper are in good agreement with theprevious investigations thus the validity of the program isverified

-e rotor unbalance mass is an important parameter forrotating machine which influence the response of the rotor-bearing system [28 29] In this paper what we focused on isthe influence of the surface waviness to the nonlinear dy-namic response So the single rotor unbalance value(015 μm) is selected in this study -e rotor-bearing systemis used in the turbomachine with a superhigh speed -usthe mass of the rotor is usually quite small less than 05 kg

In the numerical integration the aspect ratio (LD) ofself-acting journal gas bearing for calculation is chosen as144 -e lubricant in the film is air -e parameters ofelastomer are set as follows k 4e6Nm ξ 0005 and c 1To show the influence of the surface waviness to the non-linear dynamic response clearly the bifurcation diagrams ofthe rotor-bearing system with flat surface are plotted whichis presented in Figure 4 From the results we can concludethat when the half mass of the rotor is equal or less than018 kg the rotor center presents a regular periodic motionBut motion loses its regularity and becomes irregular whenthe half mass of the rotor increased to 019 kg

Figure 5 shows the bifurcation diagrams of the rotatingsystem with circumferential surface waviness In this cal-culation we assume that the surface waviness is only in-cluded in the circumferential direction of the bearing andthe number of waves is equal to 5 and the amplitude of thewaves is 1 micrometer Compared with the results in Fig-ure 4 the discrete point of the system with circumferentialwaviness is much bigger than the system with flat surfacewhich indicated that the waviness in circumferential di-rection can increase the stability of the system obviously

In order to show nonlinear dynamic performance of thesystem clearly a set of maps such as the rotor center obits

phase portraits logarithmic spectra maps in two directionsand Poincare maps are used to analyze the nonlinear dy-namic performance in different conditions

Figures 6(a)ndash13(a) are the obits of the rotor center in the (XY) plane Figures 6(b)ndash13(b) and 6(c)ndash13(c) are phase portraitsin (X _X) and (Y _Y) planes respectively Figures 6(d)ndash13(d)and 6(e)ndash13(e) are the logarithmic spectra of the non-dimensional displacement in horizontal and vertical re-spectively Figures 6(f)ndash13(f) are Poincare maps in the (X Y)plane

From Figures 6ndash8 when the half mass of the rotor is lessthan 075 kg the vibration is mainly caused by the massunbalance the frequency of rotating is much bigger thansubharmonic frequency thus the rotor center obits showa regular periodic motion When the half mass of the rotor isin the range of 075 kg and 1 kg the amplitude of sub-harmonic vibration increased and cannot be ignored -emotion losses its periodic motion and subharmonic peri-odic motion appears However the amplitude of motion isalso small When the half mass of the rotor is bigger than10 kg shown in Figures 8 and 9 the amplitude of vibrationbecomes much bigger and a closed curve occurs in thePoincare map which indicated that quasiperiodic motionappears

-e effects of the number of waviness in the circum-ferential direction on the nonlinear dynamic performance ofthe system are analyzed in which different numbers ofwaviness (N 3 5 and 7) with ms 10 kg ξ 0005 k

4e6Nm are used for calculation Figures 8ndash11 show theobits logarithmic spectrum maps and Poincare map withsurface waviness (N 3 5 and 7) respectively -e com-parison between different numbers of waviness in circum-ferential direction shows that the number of waviness do notchange the nonlinear dynamic performance of the systembut the amplitude of the vibration declines slightly with theincrease of the number of waviness (shown in the

01 02 03 04 05 06 07 08Eccentricity ratio

01 02 03 04 05 06 07 08Eccentricity ratio

Dim

ensio

nles

s for

ce

0

1

2

3

4

5

6

7

Atti

tude

angl

e(deg)

10

20

30

40

50

60

70

80

Λ = 06

Λ = 06

Λ = 30

Λ = 30

Reference [23]Reference [28]This paper

Reference [23]Reference [28]This paper

Figure 3 Comparison of steady state calculation results for journal gas bearing D L

6 Shock and Vibration

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

d2Xs

dτ2

Fex minusFgx + Msρ sin(τ)

Ms Asx

d2Ys

dτ2

Yex minusYgx + Msρ cos(τ)

Ms Asy

d2Xb

dτ2

Fgx minusKXb minus sign( _X)C _Xb1113868111386811138681113868

1113868111386811138681113868c

Mb Abx

d2Yb

dτ2

Ygx minusKYb minus sign( _Y)C _Yb1113868111386811138681113868

1113868111386811138681113868c

Mb Aby

(12)

As we get acceleration of the rotor and bearing inhorizontal and vertical directions the center obits of rotorand bearing can be acceptable by using the followingequation

Vsx Vsx0 + AsxΔτ

Vsy Vsy0 + AsyΔτ

Vbx Vbx0 + AbxΔτ

Vby Vby0 + AbyΔτ

Xs Xs0 + VsxΔτ +12AsxΔτ

2 Ys

Ys0 + VsyΔτ +12AsyΔτ

2 Xb

Xb0 + VbxΔτ +12AbxΔτ

2 Yb Yb0 + VbyΔτ +

12AbyΔτ

2

(13)

Computation of the center trajectory of rotor andbearing is an iterative procedure -e procedure begins withthe initial equilibrium state -e initial position of the rotor(X0 Y0) is a static balance position which is defined by theexternal loading and rotational speed To estimate the in-fluence of the initial condition the first 600 periodic cal-culation data can be ignored and the nonlinear behavioranalysis is based on the data after the vibration is stable -einitial velocities of the rotor in two different directions areassumed to be zero

4 Results and Discussion

In order to verify the correctness of the program the attitudeangle φ and supporting force F of a self-acting journal gasbearing with smooth surface are calculated and compared

Rotor Journal bearing

Elastomer suspension

ωOs

Ob Dde

(a)

Rotor

Elastomer suspension

Journal bearing

Disk

(b)

Figure 2 Schematic of a rigid rotor (massless shaft and disk) in journal bearing with elastic damper

Shock and Vibration 5

with previous investigations [23 28] as is shown in Figure 3-e results in this paper are in good agreement with theprevious investigations thus the validity of the program isverified

-e rotor unbalance mass is an important parameter forrotating machine which influence the response of the rotor-bearing system [28 29] In this paper what we focused on isthe influence of the surface waviness to the nonlinear dy-namic response So the single rotor unbalance value(015 μm) is selected in this study -e rotor-bearing systemis used in the turbomachine with a superhigh speed -usthe mass of the rotor is usually quite small less than 05 kg

In the numerical integration the aspect ratio (LD) ofself-acting journal gas bearing for calculation is chosen as144 -e lubricant in the film is air -e parameters ofelastomer are set as follows k 4e6Nm ξ 0005 and c 1To show the influence of the surface waviness to the non-linear dynamic response clearly the bifurcation diagrams ofthe rotor-bearing system with flat surface are plotted whichis presented in Figure 4 From the results we can concludethat when the half mass of the rotor is equal or less than018 kg the rotor center presents a regular periodic motionBut motion loses its regularity and becomes irregular whenthe half mass of the rotor increased to 019 kg

Figure 5 shows the bifurcation diagrams of the rotatingsystem with circumferential surface waviness In this cal-culation we assume that the surface waviness is only in-cluded in the circumferential direction of the bearing andthe number of waves is equal to 5 and the amplitude of thewaves is 1 micrometer Compared with the results in Fig-ure 4 the discrete point of the system with circumferentialwaviness is much bigger than the system with flat surfacewhich indicated that the waviness in circumferential di-rection can increase the stability of the system obviously

In order to show nonlinear dynamic performance of thesystem clearly a set of maps such as the rotor center obits

phase portraits logarithmic spectra maps in two directionsand Poincare maps are used to analyze the nonlinear dy-namic performance in different conditions

Figures 6(a)ndash13(a) are the obits of the rotor center in the (XY) plane Figures 6(b)ndash13(b) and 6(c)ndash13(c) are phase portraitsin (X _X) and (Y _Y) planes respectively Figures 6(d)ndash13(d)and 6(e)ndash13(e) are the logarithmic spectra of the non-dimensional displacement in horizontal and vertical re-spectively Figures 6(f)ndash13(f) are Poincare maps in the (X Y)plane

From Figures 6ndash8 when the half mass of the rotor is lessthan 075 kg the vibration is mainly caused by the massunbalance the frequency of rotating is much bigger thansubharmonic frequency thus the rotor center obits showa regular periodic motion When the half mass of the rotor isin the range of 075 kg and 1 kg the amplitude of sub-harmonic vibration increased and cannot be ignored -emotion losses its periodic motion and subharmonic peri-odic motion appears However the amplitude of motion isalso small When the half mass of the rotor is bigger than10 kg shown in Figures 8 and 9 the amplitude of vibrationbecomes much bigger and a closed curve occurs in thePoincare map which indicated that quasiperiodic motionappears

-e effects of the number of waviness in the circum-ferential direction on the nonlinear dynamic performance ofthe system are analyzed in which different numbers ofwaviness (N 3 5 and 7) with ms 10 kg ξ 0005 k

4e6Nm are used for calculation Figures 8ndash11 show theobits logarithmic spectrum maps and Poincare map withsurface waviness (N 3 5 and 7) respectively -e com-parison between different numbers of waviness in circum-ferential direction shows that the number of waviness do notchange the nonlinear dynamic performance of the systembut the amplitude of the vibration declines slightly with theincrease of the number of waviness (shown in the

01 02 03 04 05 06 07 08Eccentricity ratio

01 02 03 04 05 06 07 08Eccentricity ratio

Dim

ensio

nles

s for

ce

0

1

2

3

4

5

6

7

Atti

tude

angl

e(deg)

10

20

30

40

50

60

70

80

Λ = 06

Λ = 06

Λ = 30

Λ = 30

Reference [23]Reference [28]This paper

Reference [23]Reference [28]This paper

Figure 3 Comparison of steady state calculation results for journal gas bearing D L

6 Shock and Vibration

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

with previous investigations [23 28] as is shown in Figure 3-e results in this paper are in good agreement with theprevious investigations thus the validity of the program isverified

-e rotor unbalance mass is an important parameter forrotating machine which influence the response of the rotor-bearing system [28 29] In this paper what we focused on isthe influence of the surface waviness to the nonlinear dy-namic response So the single rotor unbalance value(015 μm) is selected in this study -e rotor-bearing systemis used in the turbomachine with a superhigh speed -usthe mass of the rotor is usually quite small less than 05 kg

In the numerical integration the aspect ratio (LD) ofself-acting journal gas bearing for calculation is chosen as144 -e lubricant in the film is air -e parameters ofelastomer are set as follows k 4e6Nm ξ 0005 and c 1To show the influence of the surface waviness to the non-linear dynamic response clearly the bifurcation diagrams ofthe rotor-bearing system with flat surface are plotted whichis presented in Figure 4 From the results we can concludethat when the half mass of the rotor is equal or less than018 kg the rotor center presents a regular periodic motionBut motion loses its regularity and becomes irregular whenthe half mass of the rotor increased to 019 kg

Figure 5 shows the bifurcation diagrams of the rotatingsystem with circumferential surface waviness In this cal-culation we assume that the surface waviness is only in-cluded in the circumferential direction of the bearing andthe number of waves is equal to 5 and the amplitude of thewaves is 1 micrometer Compared with the results in Fig-ure 4 the discrete point of the system with circumferentialwaviness is much bigger than the system with flat surfacewhich indicated that the waviness in circumferential di-rection can increase the stability of the system obviously

In order to show nonlinear dynamic performance of thesystem clearly a set of maps such as the rotor center obits

phase portraits logarithmic spectra maps in two directionsand Poincare maps are used to analyze the nonlinear dy-namic performance in different conditions

Figures 6(a)ndash13(a) are the obits of the rotor center in the (XY) plane Figures 6(b)ndash13(b) and 6(c)ndash13(c) are phase portraitsin (X _X) and (Y _Y) planes respectively Figures 6(d)ndash13(d)and 6(e)ndash13(e) are the logarithmic spectra of the non-dimensional displacement in horizontal and vertical re-spectively Figures 6(f)ndash13(f) are Poincare maps in the (X Y)plane

From Figures 6ndash8 when the half mass of the rotor is lessthan 075 kg the vibration is mainly caused by the massunbalance the frequency of rotating is much bigger thansubharmonic frequency thus the rotor center obits showa regular periodic motion When the half mass of the rotor isin the range of 075 kg and 1 kg the amplitude of sub-harmonic vibration increased and cannot be ignored -emotion losses its periodic motion and subharmonic peri-odic motion appears However the amplitude of motion isalso small When the half mass of the rotor is bigger than10 kg shown in Figures 8 and 9 the amplitude of vibrationbecomes much bigger and a closed curve occurs in thePoincare map which indicated that quasiperiodic motionappears

-e effects of the number of waviness in the circum-ferential direction on the nonlinear dynamic performance ofthe system are analyzed in which different numbers ofwaviness (N 3 5 and 7) with ms 10 kg ξ 0005 k

4e6Nm are used for calculation Figures 8ndash11 show theobits logarithmic spectrum maps and Poincare map withsurface waviness (N 3 5 and 7) respectively -e com-parison between different numbers of waviness in circum-ferential direction shows that the number of waviness do notchange the nonlinear dynamic performance of the systembut the amplitude of the vibration declines slightly with theincrease of the number of waviness (shown in the

01 02 03 04 05 06 07 08Eccentricity ratio

01 02 03 04 05 06 07 08Eccentricity ratio

Dim

ensio

nles

s for

ce

0

1

2

3

4

5

6

7

Atti

tude

angl

e(deg)

10

20

30

40

50

60

70

80

Λ = 06

Λ = 06

Λ = 30

Λ = 30

Reference [23]Reference [28]This paper

Reference [23]Reference [28]This paper

Figure 3 Comparison of steady state calculation results for journal gas bearing D L

6 Shock and Vibration

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

logarithmic spectrum maps) -e increase of the number ofwaves can only improve the static and dynamic performanceof the bearing slightly which is not enough to change thenonlinear dynamic behavior of the system

Figures 8 12 and 13 display the obits logarithm am-plitude spectrum maps and Poincare maps of the rotorcenter with different surface waviness amplitudes in cir-cumferential direction In this solution three cases (A 1 2and 3) with other parameters such as ms 1 kg ξ 0005and k 4e6Nm are used and compared -e nonlinear

dynamic performance changes from quasiperiodic motion(A 1) to regular periodic motion (A 4) -is means thelarge amplitude of the waviness can increase the stability ofthe system but the amplitude of the waviness is restricted bythe gas clearance From the obits and logarithm spectrummaps in these pictures above we can find that the increase ofthe amplitude of the surface waviness in circumferentialdirection can decline the amplitude of the subharmonicfrequency-at is because the increase of the amplitude of thewaviness can increase the static and dynamic performance

1

0

ndash1

ndash2

X s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(a)

3

2

1

0

Y s (n

T)

016 018 020 022 024 026 028 030ms (kg)

(b)

Figure 4 Bifurcation diagrams of rotor (a)Xs(nT) and (b)Ys(nT) versus half mass of rotor

ndash03

X s (n

T)

05 06 07 08ms (kg)

09 10 11 12

ndash02

ndash01

00

01

06 07 08ndash0140

ndash0135

ndash0130

(a)

11

05 06 07 08ms (kg)

Y s (n

T)

09 10 11 12

12

13

14

12306 07 08

124

125

(b)

Figure 5 Bifurcation diagrams of rotor center with circumferential surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

Shock and Vibration 7

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

minus016

minus0155

minus015

minus0145

minus014

minus0135

123

1235

124

1245

125

1255

Xs

Y s

(a)

minus016

minus0155

minus015

minus0145

minus014

minus0135

minus0015minus001minus0005

000050010015

Xs

V sx

(b)

123

1235

124

1245

125

1255minus0015

minus001minus0005

000050010015

Ys

V sy

(c)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequencyA

mpl

itude

of X

s

(d)

0 05 1 15 2 25 3 35 4minus40minus35minus30minus25minus20minus15minus10

Normalized frequency

Am

plitu

de o

f Ys

(e)

minus01

4

minus01

38

minus01

36

minus01

34

minus01

32

minus01

31221225

1231235

1241245

125

Xs (nT)

Y s (n

T)

(f )

Figure 6 Obits Poincare map and amplitude spectrum of rotor center (ms 065 kg ξ 0005 k 4e6Nm A 1μm and N 5 )

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

1225

123

1235

124

1245

125

Xs

Y s

(a)

ndash01

6

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

Xs

ndash0015

ndash001

ndash0005

0

0005

001

0015

V sx

(b)

Figure 7 Continued

8 Shock and Vibration

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

122

5

123

123

5

124

124

5

125

125

5

Ys

ndash0015

ndash001

ndash0005

0

0005

001

0015

Vsy

(c)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

38

ndash01

36

ndash01

34

ndash01

32

ndash01

3

Xs(nT)

123

1235

124

1245

125

Y s(n

T)

(f )

Figure 7 Obits Poincare map and amplitude spectrum of rotor center (ms 075 kg ξ 0005 k 4e6Nm A 1μm and N 5)

ndash01

4

ndash01

6

ndash01

5

ndash01

7

ndash01

3

ndash01

8

ndash01

2

Xs

12

122

124

126

128

13

Y s

(a)

ndash01

3

ndash01

4

ndash01

5

ndash01

2

ndash01

8

ndash01

7

ndash01

6

Xs

ndash002

ndash001

0

001

002

V sx

(b)

124

122

126 1

3

12

128

Ys

ndash002

ndash001

0

001

002

Vsy

(c)

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

05 1

15 2

250

35 43

Normalized frequency

(d)

Figure 8 Continued

Shock and Vibration 9

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

05 1 2

25

15 3 40

35

Normalized frequency

ndash30

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

1

ndash01

2

ndash01

4

ndash01

6

ndash01

5

Xs(nT)

122

124

126

128

Y s(n

T)

(f )

Figure 8 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 5)

11

12

13

14

15

Y s

ndash00

5

ndash01

5

ndash01

ndash02 0

ndash02

5

Xs

(a)

ndash004

ndash002

0

002

004

V sx

ndash01

ndash00

5 0

ndash02

ndash01

5

ndash02

5

Xs

(b)

ndash005

0

005

Vsy

1151

1

125 1

3

135 1

4

12

Ys

(c)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Xs

353 42

251

150

05

Normalized frequency

(d)

ndash25

ndash20

ndash15

ndash10

ndash5

Am

plitu

de o

f Ys

2

25 3

05 1

350 4

15

Normalized frequency

(e)

11

12

13

14

15

Y s(n

T)

ndash02

5

ndash02

ndash01

ndash00

5 0

ndash01

5

Xs(nT)

(f )

Figure 9 Obits Poincare map and amplitude spectrum of rotor center (ms 11 kg ξ 0005 k 4e6Nm A 1μm and N 5)

10 Shock and Vibration

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

Figure 14 displays the bifurcation maps of the surfacewaviness in the axial direction In this solution ms 1 kg ξ 0005 k 4e6Nm and A 1μm are used By comparingwith the flat surface bearing system we can conclude that thenonlinear dynamic performance of the system with axialsurface waviness is similar to that of the system with flatsurface -e discrete point of the system with axial surfacewaviness is 02 kg a little bigger than that of the system withflat surface (019 kg) But the existence of surface wavinessresults in the decrease of the minimum gas film which easilycauses the collision friction between rotor and bearing

5 Conclusions

In this paper the effects of surface waviness in circumfer-ential and axial direction to the nonlinear dynamic char-acteristics of the rotor-bearing system with elastomersuspension have been investigated -e nonlinear dynamic

behaviors under different conditions have been investigatedby inspecting the rotor center obits logarithm spectrummaps and Poincare maps -e results of this study are moreapplied to the practical engineering and can provide a for-ward guidance of the manufacture for the bearing

By comparing with the rotor-bearing system with flatsurface the surface waviness in circumferential directiondoes raise the stability of the system For rotor-bearingssystem with surface waviness in circumferential direction(N 5 and A 1μm) the discrete point in the bifurcationdiagram (075 kg) is much bigger than that of flat surfacesystem (018 kg)

-ree values (N 3 5 and 7) of the number of surfacewaviness are investigated and compared -e nonlineardynamic performance is insensitive to the number of thewaviness in circumferential direction With the increase ofthe number of the waviness the amplitude of the vibrationdeclines slightly

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

116

118

12

122

124

126

Xs

Y s

(a)

ndash01

6

ndash01

4

ndash01

2

ndash01

ndash00

8

ndash002

ndash001

0

001

002

Xs

V sx

(b)

116

118 1

2

122

124

126

ndash002

ndash001

0

001

002

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

ndash5

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

2

ndash01

1

ndash01

ndash00

9

ndash00

8118119

12121122123124125

Xs(nT)

Y s(n

T)

(f )

Figure 10 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 3)

Shock and Vibration 11

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

128129

13131132133134

Xs

Y s

(a)

ndash01

6

ndash01

5

ndash01

4

ndash01

3

ndash01

2

ndash01

1

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

128

129 1

3

131

132

133

134

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)0

05 1

15 2

25 3

35 4

ndash30

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Xs

Normalized frequency

(d)

0

05 1

15 2

25 3

35 4

ndash25

ndash20

ndash15

ndash10

Am

plitu

de o

f Ys

Normalized frequency

(e)

ndash01

4

ndash01

3

ndash01

2

ndash01

1

1295

13

1305

131

1315

132

Xs(nT)

Y s(n

T)

(f )

Figure 11 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 1μm and N 7)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

14

141

142

143

144

145

Xs

Y s

(a)

ndash01

55

ndash01

5

ndash01

45

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

Figure 12 Continued

12 Shock and Vibration

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

14

141

142

143

144

145

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

0

05 1

15 2

25 3

35 4ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)ndash0

14

ndash01

35

ndash01

3

ndash01

25

ndash01

2141

1415

142

1425

143

Xs(nT)Y s

(nT)

(f )

Figure 12 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 2μm and N 5)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

172

173

174

175

176

Xs

Y s

(a)

ndash01

4

ndash01

35

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash0015ndash001

ndash00050

0005001

0015

Xs

V sx

(b)

172

173

174

175

176

ndash0015ndash001

ndash00050

0005001

0015

Ys

Vsy

(c)

0

05 1

15 2

25 3

35 4

ndash35

ndash30

ndash25

ndash20

ndash15

ndash10

Normalized frequency

Am

plitu

de o

f Xs

(d)

Figure 13 Continued

Shock and Vibration 13

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

-e increase of the amplitude of the surface waviness incircumferential can improve the static and dynamic per-formance of the bearing obviously With the increase of theamplitude the motion of the system is changed fromquasiperiodic motion to regular periodic motion -us thelarge amplitude can increase the stability of the system

-e effect of waviness in the axial direction is in-vestigated by comparing with the flat surface -e axialsurface waviness cannot change the nonlinear dynamicperformance of the system But the existence of the wavinesscan reduce the minimum gas clearance which is damage forthe bearing

Nomenclatures

A Amplitude of the waves on the surface mAsx Asy Dimensionless horizontal and vertical

acceleration of rotorAbx Aby Dimensionless horizontal and vertical

acceleration of bearingCr Average gas film clearance m

d Diameter of the rotor m

eub Mass eccentricity of the rotor m

fgxfgy Supporting forces in horizontal and verticaldirections N

fex fey External forces in horizontal and verticaldirections N

fd Damping force of elastomer N

Fgx Fgy Dimensionless supporting forces in horizontaland vertical directions

F Defined asF2gx + F2

gy

1113969

φ Defined as tanminus1(FgxFgy)

Fex Fey Dimensionless external forces in horizontal andvertical directions

H Dimensionless film thickness between shaft andbearing

h Film thickness m

k c Stiffness(Nm) and damping(Nmiddotsm)of elasticdamper

c Damping exponentK C Dimensionless stiffness and damping of damperξ Dimensionless damping coefficient of elastomer

ξ c2kmb

1113968

L Length of journal bearing m

ms One-half mass of rotor Kg

mb Mass of bearing Kg

Ms Mb Dimensionless mass of rotor and bearingN Number of wavinessp Pressure Nm2

P Dimensionless pressurePa Atmosphere pressure Nm2

R Radius of shaft m

t Time s

τ Dimensionless time ϖt

U Peripheral speed of rotor in dynamical state msμ Fluid kinematic viscosity Pamiddots

0

05 1

15 2

25 3

35 4

ndash40ndash35ndash30ndash25ndash20ndash15ndash10

Normalized frequency

Am

plitu

de o

f Ys

(e)

ndash01

3

ndash01

25

ndash01

2

ndash01

15

ndash01

1173

1735

174

1745

175

Xs(nT)

Y s(n

T)

(f )

Figure 13 Obits Poincare map and amplitude spectrum of rotor center (ms 1 kg ξ 0005 k 4e6Nm A 3μm and N 5)

1

0

X s (n

T)

-1

-2

ms (kg)016 018 020 022 024 026 028 030

(a)

Y s (n

T)

1

0

ndash1

-2

ms (kg)016 018 020 022 024 026 028 030

(b)

Figure 14 Bifurcation diagrams of rotor center with axial surface waviness (a) Xs (nT) and (b) Ys (nT) versus half mass of rotor

14 Shock and Vibration

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

Vsx Vsy Dimensionless horizontal and vertical velocity ofrotor

Vbx Vby Dimensionless horizontal and vertical velocity ofbearing

ϖ Angular speed of rotor radsx y Horizontal and vertical coordinates m

Xs Ys Dimensionless horizontal and verticaldisplacement of rotor

Xb Yb Dimensionless horizontal and verticaldisplacement of bearing

Xj Yj Dimensionless displacement of rotor relative tothe bearing in horizontal and vertical directions(Xj Xs minusXb Yj Ys minusYb)

Λ Bearing numberε Eccentricity ratio of the systemδ Height distribution of waviness mλ Wavelength of surface waviness mθ η Dimensionless coordinate in circumferential and

axial directionsn Time leveli j Grid location in circumferential and axial

directionsb Parameters of bearings Parameters of rotor0 Initial parameter

Data Availability

-e data used to support the findings of this study havebeen deposited in the figshare repository (DOI 106084m9figshare6449477)

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the fund from the NationalResearch and Development Project for Key Scientific In-struments (ZDYZ2014-1) and the fund from the State KeyLaboratory of Technologies in Space Cryogenic Propellants(SKLTSCP1604)

References

[1] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoBifurcation and nonlinear dynamic analysis of a rigid rotorsupported by two-lobe noncircular gas-lubricated journalbearing systemrdquo Nonlinear Dynamics vol 61 no 4pp 783ndash802 2010

[2] R Rashidi A Karami Mohammadi and F Bakhtiari NejadldquoPreload effect on nonlinear dynamic behavior of a rigid rotorsupported by noncircular gas-lubricated journal bearingsystemsrdquo Nonlinear Dynamics vol 60 no 3 pp 231ndash2532010

[3] H E Rasheed ldquoEffect of surface waviness on the hydrody-namic lubrication of a plain cylindrical sliding elementbearingrdquo Wear vol 223 no 1ndash2 pp 1ndash6 1998

[4] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart I the wave bearing concept and a comparison tothe plain circular bearingrdquo Tribology Transactions vol 38no 1 pp 153ndash160 1995

[5] F Dimofte ldquoWave journal bearing with compressible lubri-cantmdashpart II a comparison of the wave bearing with a wave-groove bearing and a lobe bearingrdquo Tribology Transactionsvol 38 no 2 pp 364ndash372 1995

[6] Y B P Kwan and J B Post ldquoA tolerancing procedure forinherently compensated rectangular aerostatic thrust bear-ingsrdquo Tribology International vol 33 no 8 pp 581ndash585 2000

[7] T Nagaraju S C Sharma and S C Jain ldquoStudy of orificecompensated hole-entry hybrid journal bearing consideringsurface roughness and flexibility effectsrdquo Tribology In-ternational vol 39 no 7 pp 715ndash725 2006

[8] N M Ene F Dimofte and T G Keith Jr ldquoA stability analysisfor a hydrodynamic three-wave journal bearingrdquo TribologyInternational vol 41 no 5 pp 434ndash442 2007

[9] S C Sharma and P B Kushare ldquoTwo lobe non-recessedroughened hybrid journal bearing ndash a comparative studyrdquoTribology International vol 83 pp 51ndash68 2015

[10] X Wang Q Xu B Wang L Zhang H Yang and Z PengldquoEffect of surface waviness on the static performance ofaerostatic journal bearingsrdquo Tribology International vol 103pp 394ndash405 2016

[11] W A Gross and E C Zachmanaglou ldquoPerturbation solutionsfor gas-lubricating filmsrdquo Journal of Basic Engineering vol 83no 2 pp 139ndash144 1961

[12] G Belforte T Raparelli and V Viktorov ldquoModeling andidentification of gas journal bearings self-acting gas bearingresultsrdquo Yale Journal of Biology vol 124 no 4 pp 441ndash4452002

[13] L Yang S Qi and L Yu ldquoNumerical analysis on dynamiccoefficients of self-acting gas-lubricated tilting-pad journalbearingsrdquo Journal of Tribology vol 130 no 1 pp 125ndash1282008

[14] P Yu X Chen X Wang and W Jiang ldquoFrequency-dependent nonlinear dynamic stiffness of aerostatic bear-ings subjected to external perturbationsrdquo InternationalJournal of Precision Engineering and Manufacturing vol 16no 8 pp 1771ndash1777 2015

[15] Y B Kim and S T Noah ldquoBifurcation analysis for a modifiedJeffcott rotor with bearing clearancesrdquo Nonlinear Dynamicsvol 1 no 3 pp 221ndash241 1990

[16] C Wang and H Yau ldquo-eoretical analysis of high speedspindle air bearings by a hybrid numerical methodrdquo AppliedMathematics and Computation vol 217 no 5 pp 2084ndash20962010

[17] C Wang ldquo-eoretical and nonlinear behavior analysis ofa flexible rotor supported by a relative short herringbone-grooved gas journal-bearing systemrdquo Physica D NonlinearPhenomena vol 237 no 18 pp 2282ndash2295 2008

[18] C Wang M Jang and Y Yeh ldquoBifurcation and nonlineardynamic analysis of a flexible rotor supported by relative shortgas journal bearingsrdquo Chaos Solitons and Fractals vol 32no 2 pp 566ndash582 2007

[19] J Wang and C Wang ldquoNonlinear dynamic and bifurcationanalysis of short aerodynamic journal bearingsrdquo TribologyInternational vol 38 no 8 pp 740ndash748 2005

[20] C Wang and C Chen ldquoBifurcation analysis of self-acting gasjournal bearingsrdquo Journal of Tribology vol 123 no 4pp 755ndash767 2001

Shock and Vibration 15

[21] M A Hassini and M Arghir ldquoA simplified nonlinear tran-sient analysis method for gas bearingsrdquo Journal of Tribologyvol 134 no 1 article 11704 2012

[22] A Abbasi S E Khadem S Bab andM I Friswell ldquoVibrationcontrol of a rotor supported by journal bearings and anasymmetric high-static low-dynamic stiffness suspensionrdquoNonlinear Dynamics vol 85 no 1 pp 525ndash545 2016

[23] P Yang K Zhu and X Wang ldquoOn the non-linear stability ofself-acting gas journal bearingsrdquo Tribology Internationalvol 42 no 1 pp 71ndash76 2009

[24] X Wang Q Xu M Huang L Zhang and Z Peng ldquoEffects ofjournal rotation and surface waviness on the dynamic per-formance of aerostatic journal bearingsrdquo Tribology In-ternational vol 112 pp 1ndash9 2017

[25] S Yan E H Dowell and B Lin ldquoEffects of nonlineardamping suspension on nonperiodic motions of a flexiblerotor in journal bearingsrdquo Nonlinear Dynamics vol 78 no 2pp 1435ndash1450 2014

[26] G Zhang Y Sun Z Liu M Zhang and J Yan ldquoDynamiccharacteristics of self-acting gas bearingndashflexible rotor cou-pling system based on the forecasting orbit methodrdquo Non-linear Dynamics vol 69 no 1 pp 341ndash355 2012

[27] F Rudinger ldquoOptimal vibration absorber with nonlinearviscous power law damping and white noise excitationrdquoJournal of Engineering Mechanics vol 132 no 1 pp 46ndash532006

[28] R D Brown P Addison and A Chan ldquoChaos in the un-balance response of journal bearingsrdquo Nonlinear Dynamicsvol 5 no 4 pp 421ndash432 1994

[29] G Adiletta A R Guido and C Rossi ldquoChaotic motions ofa rigid rotor in short journal bearingsrdquo Nonlinear Dynamicsvol 10 no 3 pp 251ndash269 1996

16 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom