Research Article Dynamic Characteristics of the...

Research ArticleDynamic Characteristics of the Herringbone Groove GasJournal Bearings: Numerical Simulations

Bin Wang,1,2 Yongtao Sun,1,2,3,4 and Qian Ding1,2

1Department of Mechanics, Tianjin University, Tianjin 300072, China2Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin 300072, China3State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China4State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Qian Ding; [email protected]

Received 4 July 2016; Revised 17 October 2016; Accepted 15 November 2016

Academic Editor: Emiliano Mucchi

Copyright © 2016 Bin Wang et al.This is an open access article distributed under theCreativeCommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Dynamic characteristics of the herringbone grooved gas journal bearings (HGGJB) under fluid-structure interactions aresystematically investigated using the finite element method. Stability and bearing capacity of the HGGJB are estimated andcompared with those of the plain gas journal bearings (PGJB). Influences of the structural parameters, including the spiral angle,the groove number, the groove depth, the pressure relief hole diameter, the bearing radial clearance, the length to diameter ratio,and the rotating speed, on dynamic characteristics of HGGJB are analyzed. To verify the numerical simulation results, pressurenephograms and cross-section pressure curves of the same rotor model, calculated by the numerical simulation and the theoreticalmethod, respectively, are compared. Similar results are obtained. Compared to the common constrained boundary conditions inthe previous numerical simulations, boundary conditions adopted in this paper are complete self-absorption and the change of thegas inlet and outlet depends on the rotating state of the rotor, which are more accordant with the real dynamic characteristics ofthe HGGJB. In all, the results presented in this paper provide a deeper and better understanding of the dynamic characteristics ofthe HGGJB under fluid-structure interactions.

1. Introduction

As a vital part of the rotor system, the herringbone groovedgas journal bearings (HGGJB) have important applications inhigh speed, rotatingmachinery (such as those found in textilemachinery, refrigerator, gas recycling system, and so on) dueto their attractive stability characteristics. This advantage isvery important for journal concentric operation within thebearing annular space since herringbone grooves can inhibitthe whirl instability which often occurs in the self-actingjournal bearings [1].

To deeply and completely understand the dynamic char-acteristics of the HGGJB, a lot of research has been done.Zirkelback and San Andres [1] introduced the finite elementanalysis for prediction of the static and dynamic responses ofa rotor supported by HGGJBs. The optimum dynamic coeffi-cientswere determined through analysis of the bearing geom-etry parametric. Jang and Chang [2] investigated the effects

of cavitation on the performance of a herringbone groovedjournal bearing of the spindle motor in a computer hard diskdrive. The Reynolds equation incorporating Elrod algorithmwas transformed to computational space and solved by thefinite volume method. Jang and Yoon [3] performed thenonlinear dynamic analysis of a herringbone grooved journalbearing with plain sleeve and a plain journal bearing withherringbone grooved sleeve under static and dynamic load,where FEM is used to solve the Reynolds equation in orderto calculate the pressure distribution in a fluid film. Rao andSawicki [4] analyzed stability characteristics of the herring-bone grooved journal bearings taking into account cavitationof the fluid flow. Based on the experimental method, Gad etal. [5] introduced the beveled-step groove profile to obtain theoptimum groove geometry for herringbone grooved journalbearings. The group of Cheng-Chi Wang performed bifur-cation analysis of an aerodynamic journal bearing systemconsidering the effect of stationary herringbone grooves [6].

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 8743016, 13 pageshttp://dx.doi.org/10.1155/2016/8743016

2 Shock and Vibration

They also theoretically studied the nonlinear behavior ofthe flexible rotor supported by HGGJB [7], a relative shortHGGJB system [8], and the spherical gas-lubricated bearingsystem [9], respectively. Hirayama et al. [10] discussed theoptimization of groove dimensions in herringbone-groovedjournal bearings with the aim of designing precision spindleswith improved run-out characteristics. Based on the useof frequency dependent dynamic coefficients and on therational function approximation of the resulting impedances,Hassini and Arghir proposed a novel simplified method forsimulating large nonlinear displacements [11] and stabilityanalysis in gas lubricated bearings [12, 13].Thismethod avoidssolving the unsteady compressible thin film fluid equations,can be applied to any bearing or seal whose impedance isapproximated by rational transfer functions, and is applicablefor both rigid and flexible rotors. Schiffmann and Favrat [14]compared the influences of the real gas and the perfect gas onherringbone grooved journal bearings and encouraged bear-ing designers to carefully consider the real gas effects. Chen etal. [15] analyzed the HGGJB with groove-ridge discontinuityand obtained the distribution of the pressure in the fluidfilm. Schiffmann [16] found the optimum groove geometryby coupling the gas bearing supported rotor model with amultiobjective optimizer and attempted to set up generaldesign guidelines for enhanced gas-lubricated herringbonegrooved journal bearings.

However, most of the above-mentioned researches inves-tigated either the static characteristics and the dynamic coef-ficients of HGGJB in steady state or the optimization of onlyone parameter of the HGGJB or a new method for stabilityanalysis of the HGGJB. A systematic study about the influ-ences of all the related parameters on dynamic characteristicsof theHGGJB has seldom been done.Therefore, in this paper,through the commercial software Ansys�, the dynamic char-acteristics of the HGGJB under fluid-structure interactionsare systematically studied. First, to verify the correctness ofthe numerical simulations carried out in this paper, the pres-sure nephograms and the cross-section pressure curves of thesame rotor model, calculated by the theoretical method andthe numerical simulation, respectively, are compared. Then,to highlight advantages of the HGGJB, the stability and bear-ing capacity of the HGGJB are compared with those of theplain gas journal bearings (PGJB). Finally, the effects of dif-ferent parameters, including the spiral angle, the groove num-ber, the groove depth, the pressure relief hole diameter, thebearing radial clearance, and the length to diameter ratio, ondynamic characteristics of the HGGJB are given one by one.At the same time, effect of the bearing capacity on rotor tra-jectory of the HGGJB is shown.

In the previous numerical simulations on dynamic char-acteristics of theHGGJB, the boundary conditions are usuallyconstrained. In this paper the boundary conditions of theHGGJB are complete self-absorption and the change of thegas inlet and outlet depends on the rotating state of the rotor,which confirm better to the real dynamic characteristics ofthe HGGJB.

Table 1: Basic parameters of the journal bearing rotor system.

Diameter of thedisk𝐷/(mm) Axial length/diameter𝐿0/𝐷0/(mm)

Thickness/diameter ofthe bearing pedestal𝐵1/𝐷1/(mm)

100 200/50 20/50.10

Figure 1:Thegeometrymodel of the journal bearing rotor structure.

2. The Geometric Model of the Rotor Systemand the Finite Element Model

First, the geometrical model of a rotor system and the finiteelement model used for analyzing the rotor system underfluid-structure interactions are described.The details of fluid-structure interactions are given in Appendix A.

2.1. The Geometric Model of the Rotor Structure. Figure 1shows the typical geometry of a journal bearing rotor struc-ture, in which the gas film can be plain type (Figure 2(a)) orherringbone groove type (Figure 2(b)). A compressible gasfills the space between the stationary bearing and the rotatingjournal. The fixed basic parameters of the rotor system aregiven in Table 1.The width 𝐵0 of the disk varies with the bear-ing capacity and the length 𝐿1 of the bearing pedestal varieswith the length to width ratio of the bearing. The modelsgiven in Figures 1 and 2 are used to analyze the fluid-structureinteractions of the rotor bearing system through the FEM.

2.2. The Finite Element Model. Figure 3 depicts the mesh ofthe journal bearing rotor structure. Local grids of the plainand the herringbone groove gas films are shown in Figures4(a) and 4(b), respectively. Both the rotor structure and thegas film are divided into hexahedral grids. Compared to theunstructured tetrahedral grid, the hexahedral grid structurehas higher calculation accuracy and better quality of grid,especially for the gas film clearance flow problem. It cangreatly reduce the number of grids and improve the calcu-lation efficiency at the same precision.

To describe rotor’s trajectory, the dynamic grid technol-ogy is used. The boundaries are the opening inlet and outlet.The initial position is the centre of the bearing pedestal. Inprocess of the fluid-structure interactions, the bearing gasfilm deforms due to the load from the rotor (Figures 5 and 6).The grid deformation could reflect the dynamic performanceof the rotor system during the calculation process. Dynamicgrid technique could completely describe the rule of therotor-bearing system.

Shock and Vibration 3

(a) (b)

Figure 2: (a) The plain gas film model; (b) the herringbone groove gas film model.

Figure 3: The finite element model of the rotor system.

Table 2: The change of the eccentricity with the grid number.

Grid number 2165428 4279624 8026475 15897792 30879884Eccentricity (𝜇m) 47.31 44.28 42.56 41.60 41.59

2.3. The Grid Sensitivity Analysis. In order to eliminate theinfluence of the grid number, the grid sensitivity is analyzed(Table 2). The parameters of the model are the optimizedparameters whichwill be discussed in Section 3. FromTable 2it is easy to see that the eccentricity decreases with theincrease of the grid number and it is almost the same whenthe grid number increases from 15897792 to 30879884. Inorder to increase the calculation efficiency, 15897792 isselected for the grid number.

3. Verification of the Numerical Simulations

The pressure nephogram reflects the bearing pressure dis-tribution under certain bearing capacity and rotating speed.And the cross-section pressure curves can be more intuitiveto reflect the rule of circumferential pressure distribution. Toverify the correctness of the numerical simulations carriedout in this paper, the pressure nephograms and the cross-section pressure curves of the same rotor model for theHGGJB and PGJB, obtained, respectively, by the theoreticalmethod and the numerical simulation, are compared in thispart. The parameters for the HGGJB are the following opti-mized parameters: groove number 12, spiral angle number

Table 3: Basic parameters of the PGJB.

𝐿 (mm) 𝑅 (mm) 𝐶0 (mm) 𝑃𝑎 (Pa) 𝜇 (Pa ⋅ s) 𝜔 (r/min) Λ60 25 0.05 101325 1.8 × 10−5 1.0 × 104 1.7

45∘, groove depth 50 𝜇m, pressure relief hole diameter 3mm,bearing radial clearance 50 𝜇m, and length to diameter ratio1.2. In addition, the recess width ratio is fixed and it is 1 : 1.The rotating speed of the rotor and the bearing capacity arekept at 10000 r/min and 2 kg.The parameters of the PGJB aregiven in Table 3, and the bearing capacity is 0.5 kg.

To theoretically analyze the pressure of the PGJB andthe HGGJB, usually the method of the similarity theory isadopted to derive the dimensionless form of the Reynoldsequation and then solve the Reynolds equation through thefinite difference method in which the Reynolds equation istransferred from the rectangular coordinate system to theoblique coordinate system.The details of the finite differencemethod are given in Appendix B.

The dimensionless Reynolds equation for the PGJB in therectangular coordinate system is

𝜕𝜕𝜑 (𝐻3 𝜕𝑝𝜕𝜑) + (

𝐷𝐿2) 𝜕

𝜕𝜆 (𝐻3𝜕𝑃𝜕𝜆) = 3𝜕𝐻𝜕𝜑 , (1)

where𝐻 = 1 + 𝜀 cos𝜑 (𝜀 = 𝑒/𝐶0), 𝜆 = 2𝑧/𝐿 (𝐿 is the lengthof the bearing),𝐻 = ℎ/𝐶0 (𝐶0 is the clearance of the radius),and 𝑃 = 𝜓2𝑝/2Ω𝜇 (𝜓 = 𝐶0/𝑅 is the ratio of the clearance).

Through the following coordinate transformation (Fig-ure 7):

𝜑 = 𝑥1 + 𝑦1 sin𝛽,𝜆 = 𝑦1 cos𝛽 (2)


(a) (b)

Figure 4: The local grid of (a) the plain gas film and (b) the herringbone groove gas film.

0

0.0001

0.0002

0.0003

0.0004

(m)

0

0.0001

0.0002

0.0003

0.0004

(m)

Figure 5: Grid deformation of the plain gas film.

the Reynolds equation is transferred from the rectangularcoordinate system (in (1)) to the oblique coordinate systemfor HGGJB as

𝜕𝜕𝑥1 (𝐻

3 𝜕𝑝𝜕𝑥1) + (𝐷𝐿 )2

⋅ 1cos𝛽 { 𝜕

𝜕𝑦1 [𝐻3cos𝛽 ( 𝜕𝑝

𝜕𝑦1 −𝜕𝑝𝜕𝑥1 sin𝛽)]

− 𝜕𝜕𝑥1 [

𝐻3cos𝛽 ( 𝜕𝑝

𝜕𝑦1 −𝜕𝑝𝜕𝑥1 sin𝛽)] sin𝛽} = 3 𝜕𝐻𝜕𝑥1 ,

(3)

in which the gas film thickness 𝐻 under the oblique coordi-nate system is

𝐻 = 1 + 𝜀 cos (𝑥1 + 𝑦1 sin𝛽 − 𝜃) . (4)

Discretizing the bearing lubrication film and solving theReynolds Eqs. (1) and (3) through the finite differencemethodgives the pressure nephogram of the rotor system and thecorresponding circumferential pressure distribution curveof the cross-section for the PGJB (Figure 8) and HGGJB(Figure 9), respectively. In the calculation, the environmental,

0

0.0001

0.0002

0.0003

0.0004

(m)

0

0.0001

0.0002

0.0003

0.0004

(m)

Figure 6: Grid deformation of the herringbone groove gas film.

symmetric, and periodic boundary conditions for the finitedifference method are, respectively, as follows:

𝑃( 𝑙2𝑟 , 𝜑) = 1, (5)

𝑃 (𝜆, 𝜑) = 𝑃 (−𝜆, 𝜑) = 1 atm, (6)

𝑃 (𝜆, 𝜑) = 𝑃 (𝜆, 𝜑 + 2𝜋) . (7)

Comparing the pressure nephograms by the theoreticalmethod (Figures 8(a) and 9(a)) [2] and the numerical simu-lation (Figures 8(b) and 9(b)) [17], respectively, it is easy toreveal that the pressure nephogram distribution changingrule is similar. The maximum pressure peak appears in theposition of the largest eccentricity. Positive and negativepressures distribute along the circumference. Comparing thepressure nephograms of Figures 8 and 9, respectively, it iseasy to find that the pressure of the HGGJB is more than thePGJB.The reason is that theHGGJB havemore stable bearingcapability than the PGJB. Also the change rule of the cross-section pressure values (Figures 8(c) and 9(c)) is similar ona whole, respectively, and all of them have a pressure peakin the biggest eccentric the same as the pressure nephogram.Besides, the trend of Figures 8(c) and 9(c) is like Figure 6(b)and Figure 7(b), respectively, in [15].

To sum up, for the PGJB and HGGJB on the whole boththe pressure nephograms and the cross-section pressure


𝜆 y1

𝛽

x1 𝜑

P(𝜑, 𝜆)

Figure 7: The oblique coordinate transformation.

curves calculated by the theoretical method and the numer-ical simulation are similar, respectively, illustrating correct-ness of our numerical simulations.

4. Analysis of the Calculation Results

In Sections 4.1 and 4.2, the parameters used for the HGGJBand PGJB are the parameters mentioned at the beginning ofSection 3. The widths 𝐵0 of the disk corresponding to thebearing capacities 0.5 kg, 1 kg, 2 kg, and 2.5 kg are 7.5mm,15mm, 30mm, and 37.5mm, respectively. And the lengths𝐿1 of the bearing pedestal corresponding the length to widthratios of the bearing capacities 0.8, 1.0, 1.2, and 1.6 are 40mm,50mm, 60mm, and 80mm, respectively. In Section 4.3,influence of each parameter on dynamic characteristics of theHGGJB is analyzed, respectively, which is the only studiedparameter that varied, whilst the other parameters keep theabove-mentioned values.

4.1. Stability of the HGGJB Comparing with the PGJB. Tohighlight the advantage of the HGGJB in stability comparingwith that of the PGJB, the rotor trajectory and the polar coor-dinates of the PGJB (Figure 10) and the HGGJB (Figure 11) atthe rotating speed of 10000 r/min and under different bearingcapacities,𝑀 = 1 kg, 2 kg, and 2.5 kg, are presented.Thepolarcoordinate indicates that position of the rotor centre is atsteady state. One finds that with the increase of the load bear-ing capacity the rotor eccentricity increases but the deviationangle decreases.

Basic operating parameters of the PGJB, including lengthL, radius 𝑅 and rotating speed 𝜔 of the gas bearing, clearanceof the gas film 𝐶0, pressure of the atmosphere 𝑃𝑎, viscosity ofthe air 𝜇, and the number of bearing, are shown in Table 3.Basic operating parameters of the HGGJB are already intro-duced at the beginning of Section 3; here we do not repeatthem anymore and we also will not mention them anymorein the following parts. The initial condition is that the rotorbegins to rotate from the center of the bearing under thestandard atmospheric pressure.

From the trajectories shown in Figures 10(a) and 11(a),with the same rotating speed and the bearing capacity, wefind that when the PGJB rotor operates in periodic whirling

motion, the HGGJB rotor can settle in an equilibrium posi-tion all the time. So the HGGJB has higher stability. Thereason is that the existence of herringbone grooves makes thesucked gas flow to the middle position of the agglomeration,increasing the stability of the flow. At the same time, theexistence of the herringbone grooves inhibits the regularperiodic whirling motion of the shaft center, making theHGGJB rotor more stable.

Figure 10(a) also shows that with the increase of the bear-ing capacity the rotating inertia increases, which makes theperiodic whirling motion trajectory of the PGJB ovate fromelliptic and the periodic whirling motion range becomesbigger and bigger.

The rotor trajectory of the HGGJB (Figure 11(a)) revealsobviously that the stability is becoming stronger with theincrease of the bearing capacity and the time needed to reachthe steady state is becoming shorter. On the other hand, thestability is becoming weaker with the decrease of the bearingcapacity and the rotor trajectory tends to be unstable, whichimplies that the HGGJB is instable at the light load.

Further, we compare the polar coordinates of the HGGJB(Figure 11(b)) with that of the PGJB (Figure 10(b)), whichreflect the change of the eccentricity and attitude angle of therotor centre under different bearing capacities. Apparentlythe periodic whirling motion center of the PGJB attitudeangle is relatively larger, which has more than 90 degreesunder the light load. This indicates that the PGJB is in theunsteady periodic whirling motion state, while the attitudeangle of the HGGJB (Figure 11(b)) is less than 90 degrees,which means that the HGGJB rotor is in a stable transientrotating motion.

To sum up, under the same rotating speed and bearingcapacity, the HGGJB is more stable than the PGJB.

4.2. Bearing Capacity of the HGGJB Compared with the PGJB.To further highlight the advantage of theHGGJB, in this part,the maximum stable loading capacity of the HGGJB is com-pared with that of the PGJB. For this purpose, the rotor tra-jectory and polar coordinates of the HGGJB and PGJB underthe same rotating speed, 10000 r/min, are given in Figures12(a) and 9(b), respectively. Here, basic parameters of thePGJB are still those shown in Table 3 and the bearing capa-cities for the PGJB and HGGJB are 0.5 kg and 2 kg, respec-tively.

From Figure 12(a) we can see that the herringbone groovebearing moves quickly to achieve a stable equilibrium stateafter a few spiral laps from an origin position. For the plainbearing with poor flow regularity, it needs more spiral lapsfrom the origin of coordinates to the equilibrium position.The bearing capacity for the PGJB is 0.5 kg in this case.When the bearing capacity is increased, the plain bearing willbecome unstable and the bearing periodic whirling motionphenomenon will appear.

From the polar coordinates of Figure 12(b), it is apparentthat the stable center of the plain bearing is more near tothe center of the bearing, implying that the plain bearing ismore sensitive to the external disturbance excitation, whilethe herringbone groove bearing center is far away from thebearing center, meaning that theHGGJB can runmore stably.


𝜆

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−10 1 2 3 4 5 6 7

𝜑

(a)

Pressure1.2

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

−0.4

−0.6

−0.80

0.0075

0.015

0.0225

0.03

(m)

(b)

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

1.5

p

Finite difference methodFinite element method

𝜑

(c)

Figure 8: Nondimensional pressure nephogram of the PGJB by (a) the finite difference method and (b) the finite element method; (c) thecorresponding circumferential pressure distribution curve of the cross-section of PGJB by the finite difference method and the finite elementmethod.

In conclusion, both comparisons of the rotor trajectoryand polar coordinates illustrate that under the same rotatingspeed the stable bearing capacity of the HGGJB is larger(2 kg/0.5 kg = 4 times) than that of the PGJB.

4.3. Influences of the Structural and Operation Parameters onDynamic Characteristics of the HGGJB

4.3.1. The Spiral Angle. First, the effect of the spiral angle ondynamic characteristics of the HGGJB is analyzed. To thisend, rotor trajectory and polar coordinates of the HGGJBunder different spiral angle (𝛽 = 30∘, 45∘, 60∘, 90∘) are calcu-lated, as shown in Figure 13.

From Figure 13(a), we can see that with the increase ofthe spiral angle, the transient fluctuation decreases first thenincreases and the final equilibrium position migrates down-ward along the 𝑦-axis. When the spiral angle reaches 90∘ theperiodic whirling motion appears. This indicates that the air

collection of the constraint function is becoming strongerthen weaker with the spiral angle increasing.

At the same time, the polar coordinates of Figure 13(b)show that with the increase of the spiral angle, the eccentricitydecreases, and the attitude angle is reduced accordingly. It isconsistent with the rotor trajectory. When the spiral angle is45∘, the equilibrium position is (17.74, 37.64) and the HGGJBacquires higher stability compared with the spiral angles 30∘,60∘, and 90∘ cases. When the spiral angle is 90∘, the periodicwhirlingmotion center elevates. Because the rotor operates ina small periodic whirling motion finally for both the 30∘ and60∘ cases and this phenomenon is more obvious for the 90∘case, it indicates that the spiral angle should be neither toobig nor too small.

4.3.2. The Groove Number. In this part, the effect of thegroove number on dynamic characteristics of the HGGJB isstudied. For this purpose, the rotor trajectory and polar


𝜆

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

0.8

1

1.2

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−1−1 0 1 2 3 4 5 6 7

𝜑

(a)

Pressure1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

−0.2

−0.4

−0.6

0.015

0.030

(m)

(b)

0 1 2 3 4 5 6 7−0.6−0.4−0.2

00.20.40.60.8

11.21.4

p

Finite difference methodFinite element method

𝜑

(c)

Figure 9: Nondimensional pressure nephogram of the HGGJB by (a) the finite difference method and (b) the finite element method; (c)the corresponding circumferential pressure distribution curve of the cross-section of HGGJB by the finite difference method and the finiteelement method.

0 5 10 15 20 25 30 35 40

−30

−20

−10

0

10

20

30

x (𝜇m)

y(𝜇

m)

M = 1kgM = 2kgM = 2.5 kg

(a)

10

20

30

30

210

60

240

90 270

120

300

150

330

180

0

(104.0, 16.3)(97.7, 24.5)(82.5, 27.4)

M = 1kgM = 2kgM = 2.5 kg

r (𝜇m), 𝜃 (∘)

(b)

Figure 10: (a) Rotor trajectory and (b) polar coordinates of the PGJB under different bearing capacities at the rotating speed of 10000 r/min.


0 5 10 15 20 25 30 35 40−40−35−30−25−20−15−10

−50

M = 1kgM = 2kgM = 2.5 kg

x (𝜇m)

y(𝜇

m)

(a)

30

210

60

240

90 270

120

300

150

330

180

0

(79.5, 28.8)

(64.8, 41.6)(56.8, 45.7)

1020304050

M = 1kgM = 2kgM = 2.5 kg

r (𝜇m), 𝜃 (∘)

(b)

Figure 11: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different bearing capacities at the rotating speed of 10000 r/min.

0 5 10 15 20 25 30 35 40−40−35−30−25−20−15−10

−505

10

PGJBHGGJB

x (𝜇m)

y(𝜇

m)

(a)

PGJBHGGJB

30

210

60

240

90 270

120

300

150

330

180

0

(89.5, 24.9)

(64.8, 41.6)

r (𝜇m), 𝜃 (∘)

10203040

50

(b)

Figure 12: (a) Rotor trajectory and (b) polar coordinates of the HGGJB and PGJB at the rotating speed of 10000 r/min.

0 5 10 15 20 25 30 35 40 45−40−35−30−25−20−15−10

−505

x (𝜇m)

y(𝜇

m)

𝛽 = 30∘

𝛽 = 45∘𝛽 = 60∘

𝛽 = 90∘

(a)

1020304050

30

210

60

240

90 270

120

300

150

330

180

0

(69.2, 42.6)(64.8, 41.6) (60.4, 38.7)

(68.3, 41.3)

r (𝜇m), 𝜃 (∘)

𝛽 = 30∘

𝛽 = 45∘𝛽 = 60∘

𝛽 = 90∘

(b)

Figure 13: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different spiral angles at the rotating speed of 10000 r/min.


0 5 10 15 20 25 30 35 40−40−35−30−25−20−15−10

−50

x (𝜇m)

y(𝜇

m)

N = 10

N = 11

N = 12

N = 14

(a)

1020304050

30

210

60

240

90 270

120

300

150

330

180

0

(70.6, 38.9) (73.2, 39.2)(64.8, 41.6) (64.0, 37.9)

r (𝜇m), 𝜃 (∘)

N = 10

N = 11

N = 12

N = 14

(b)

Figure 14: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different groove numbers at the rotating speed of 10000 r/min.

0 5 10 15 20 25 30 35 40−40

−30

−20

−10

0

10

20

x (𝜇m)

y(𝜇

m)

H1 = 25𝜇mH1 = 50𝜇mH1 = 75𝜇m

(a)

1020304050

30

210

60

240

90 270

120

300

150

330

180

0

(83.0, 33.0)

(64.8, 41.6)

(38.4, 42.2)

r (𝜇m), 𝜃 (∘)

H1 = 25𝜇mH1 = 50𝜇mH1 = 75𝜇m

(b)

Figure 15: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different groove depth at the rotating speed of 10000 r/min.

coordinates of the HGGJB under different groove numbers𝑁 = 10, 11, 12, 14 are depicted in Figure 14.From Figure 14 it is easy to see that when the groove

number is less than or larger than 12, the transient rotatingmotion of the rotor is unstable in a region as it approachesthe equilibriumposition. In fact, the rotor cannot settle at a fixequilibrium point but whirls around a small area eventuallyif the groove number is less than or larger than 12.

With the groove number increasing from 10 to 12, the fluidis becomingmore regular and, when it is 14, the fluid becomesirregular. With the groove number increasing from 10 to 12,the eccentricity and attitude angle increase and they decreasewhen the groove number is 14.

4.3.3. The Groove Depth. To reveal the influence of thegroove depth on dynamic characteristics of the HGGJB,rotor trajectory and polar coordinates of the HGGJB underdifferent groove depths𝐻1 = 25, 50, and 75 𝜇mare depicted inFigure 15. One finds that when the groove depth is 25𝜇m, the

rotor settles at a periodic whirlingmotion with theminimumeccentricity and the largest attitude angle. When the groovedepth is 75𝜇m, the transient rotating motion is unstable,whereas it approaches to transient rotating motion at thefixed equilibrium position finally with the large eccentricityand low attitude angle. When the groove depth is 50𝜇m, theeccentricity and attitude angle are between the above twokinds of states.

4.3.4. The Pressure Relief Hole Diameter. To analyze theinfluence of the pressure relief hole diameter [18] on dynamiccharacteristics of the HGGJB, rotor trajectory and polarcoordinates of theHGGJB under different pressure relief holediameter 𝐷2 = 0, 2, 3mm are given in Figure 16. Obviously,the rotational attitude angle and the eccentricity stabilitygradually increase with the increase of the pressure relief holediameter. Specially, when the pressure relief hole diameter is0mm, the bearing outlets become the air inlet and outlet atthe same time, which leads to the bearing gases colliding each


0 5 10 15 20 25 30 35 40−40−35−30−25−20−15−10

−50

x (𝜇m)

y(𝜇

m)

D2 = 0mmD2 = 2mmD2 = 3mm

(a)

1020304050

30

210

60

240

90 270

120

300

150

330

180

0

(57.4, 39.9)(60.8, 41.9)(64.8, 41.6)

r (𝜇m), 𝜃 (∘)

D2 = 0mmD2 = 2mmD2 = 3mm

(b)

Figure 16: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different pressure relief hole diameter at the rotating speed of10000 r/min.

0 10 20 30 40 50 60 70−60−50−40−30−20−10

01020

x (𝜇m)

y(𝜇

m)

C = 25𝜇mC = 50𝜇mC = 75𝜇m

(a)

20

40

60

30

210

60

240

90 270

120

300

150

330

180

0

(52.9, 14.6)(64.8, 41.6)

(91.7, 54.4)

r (𝜇m), 𝜃 (∘)

C = 25𝜇mC = 50𝜇mC = 75𝜇m

(b)

Figure 17: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different bearing radial clearance at the rotating speed of10000 r/min.

other in the process of the flow, thus producing internal self-excited vibration and affecting the movement of the rotor.With the pressure relief hole diameter increasing from 0mmto 2mm, the eccentricity and the attitude angle increaseobviously; correspondingly the stability increases. With thepressure relief hole diameter increasing from 2mm to 3mm,the eccentricity changes a little but the attitude angle increasessignificantly, still meaning that the stability of the rotatingrotor is increased. Apparently, the bigger the pressure reliefhole diameter, the better the performance of the HGGJB.This indicates that the significance of the pressure relief holediameter on the performance of HGGJB is to regulate the gassuction and pump out and restrain the motion of the gas inthe middle of the HGGJB and thus make the rotor operatemore stably.

4.3.5. The Bearing Radial Clearance. Influence of the bearingradial clearance 𝐶 on dynamic characteristics of the HGGJBis revealed by Figure 17 for 𝐶 = 25, 50, and 75 𝜇m. One findsthat when the bearing radial clearance is 25𝜇m, the eccentric-ity and attitude angle are relatively small, and the transientfluctuation is also slight. When the bearing radial clearanceincreases to 50𝜇m, the rotor is transient rotating motion inthe stable position,whereas the attitude angle and eccentricityincrease accordingly. As the film thickness increases to 75𝜇m,the rotor motion becomes unstable in the last stage. Theattitude angle and eccentricity increase continuously andthe former reaches more than 90 degrees eventually. So forthe examples studied above, 50 𝜇m is the optimized bearingradial clearance. The result indicates that the bearing radialclearance should be the same as the groove depth.


0 5 10 15 20 25 30 35 40−50

−40

−30

−20

−10

0

10

x (𝜇m)

y(𝜇

m)

L/D = 0.8

L/D = 1L/D = 1.2

L/D = 1.6

(a)

10203040

50

30

210

60

240

90 270

120

300

150

330

180

0

(35.4, 45.3)(50.3, 42.3)

(64.8, 41.6)

(80. 4, 23.7)

r (𝜇m), 𝜃 (∘)

L/D = 0.8

L/D = 1L/D = 1.2

L/D = 1.6

(b)

Figure 18: (a) Rotor trajectory and (b) polar coordinates of the HGGJB under different length to diameter ratio at the rotating speed of10000 r/min.

4.3.6. The Length to Diameter Ratio. Influence of the lengthto diameter ratio 𝐿/𝐷 on dynamic characteristics of theHGGJB is revealed by Figure 18 for 𝐿/𝐷 = 0.8, 1, 1.2, 1.6. It isobvious that the length to diameter ratio has great influenceon dynamic characteristics of the HGGJB. With the increaseof the length to diameter ratio, the stability of the rotatingrotor is increased. Whilst the attitude angle increases, theeccentricity has experienced a process of decreasing. Theeccentricity and the attitude angle reflect the bearing capacity.The bigger the attitude angle and the smaller the eccentricity,the more improvement of the rotating change space of therotor. So the maximum bearing capacity will increase withthe attitude angle increasing and the smaller the eccentricitydecreasing.Therefore, increasing the length to diameter ratiocan effectively improve the bearing capacity.

5. Conclusions

In this paper, dynamic characteristic of the HGGJB underfluid-structure interactions are systematically studied usingCFD and FEMmethod. Some conclusions are as followings:

(1) Under the same rotating speed and bearing capacity,the HGGJB have higher stability ability than thePGJB.

(2) The spiral angle, the film thickness, the groove num-ber and the groove depth should be neither too bignor too small. The bigger the pressure relief holediameter and the length to diameter ratio, the morestable the HGGJB.

(3) Due to the constraints of the herringbone grooves, theair flows from the herringbone groove to the middlepart of the bearing which makes the air distributionmore regular and thus increases the stability of therotor system. This is the fundamental reason why theHGGJB are superior.

NodeControl surface

IP face

Zero distanceSending side Receiving side

Figure 19: Schematic diagram of the nodes coupling in fluid-structure interactions.

Appendix

A. Basic Principle of the FluidStructure Interactions

To simulate the dynamic characteristics of the HGGJB andPGJB through Ansys, the fluid structure interactions areinvolved. It means that in the calculation the fluid and thestructure need to transfer data to each other.The basic princi-ple of the fluid structure interactions, that is, the data transferprocess of the nodes coupling, is depicted in Figure 19. Inorder to transfer the data between the nonsimilar grids, the


123j

j

1 2 3 i

i

1

𝜆

−1 n + 1

...

...

· · · · · · m m + 1

𝜑

𝜑 = 𝜑1 𝜑 = 𝜑2

Figure 20: Schematic diagram of the expanded grids of the bearing.

nodes on each grid must be mapped to the correspondinggrid cell. To transfer displacements, the nodes of the fluidcoupling surface must be mapped to the solid coupling cell.To transfer the stress, the nodes of the solid coupling cellmustbe mapped to the fluid coupling surface. That is to say, forone time of node coupling, two times of data transfer mustbe implemented.

B. Basic Principle of the FiniteDifference Method

To calculate pressure distribution of the gas bearing by thefinite difference method, the gas film needs to be divided intomany grids. For a cylindrical bearing, the outer surface isexpanded into a rectangular face along the generatrix andthen evenly meshed in 𝜑 and 𝜆 directions, as shown inFigure 20.

The mesh nodes are numbered orderly according to therow and column they belong to. The numbers are designatedas 𝑖 (𝑖 = 1, . . . , 𝑚 + 1) and 𝑗 (𝑗 = 1, . . . , 𝑛 + 1), and then thespace steps are Δ𝜑 = (𝜑2 −𝜑1)/𝑚 and Δ𝜆 = 𝐿/𝑛, respectively.The pressure at node (𝑖, 𝑗) is 𝑃𝑖𝑗.

For herringbone bearing structure, after expansion, itssurface is segmented by grooves. Parallel quadrilateral meshalong the groove is then used in the computation as shown inFigure 21. Areas 𝑎 and 𝑏 contain the nodes only in and out ofthe groove, respectively, whereas areas 𝑐 and 𝑑 contain bothof them.

Nomenclature

𝐷0: Diameter of the disk (mm)𝐵0: Width of the disk (mm)𝐿0: Axial length (mm)𝐷: Axial diameter (mm)𝐵1: Thickness of the bearing pedestal (mm)𝐷1: Diameter of the bearing pedestal (mm)𝐿1: Length of the bearing pedestal (mm)

a b

c

d

Groovey1

x1

Figure 21: Schematic diagram of the grids distribution underoblique coordinate system.

𝐿: Length of the gas bearing (mm)𝑅: Radius of gas bearing (mm)𝜔: Rotating speed of gas bearing (rad/s)𝐶0: Clearance of gas film (mm)𝑃𝑎: Atmosphere𝜇: Viscosity of air (Pa⋅s)𝛽: Spiral angle𝑁: Groove number𝐻1: Groove depth (mm)𝐷2: Pressure relief hole (mm)𝐶: Bearing radial clearance (mm)𝐿/𝐷: Length to diameter𝜀: Eccentricity ratio𝑒: Eccentric distance (mm)𝜓: The ratio of the clearance𝜑: Angle𝑥1, 𝑦1: Oblique coordinate𝑧: Cartesian coordinate𝑃: Pressure (Pa)ℎ: Circumferential clearance (mm)𝜆: Dimensionless axial distance𝐻: Dimensionless circumferential clearance.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The research is supported by the National Natural Scienceof China under Grants nos. 51575378 and 11502162 andthe NSF of Tianjin under Grants nos. 13JCZDJC34900 and15JCQNJC05100. Yongtao Sun is also supported by the SeedFoundation of Tianjin University (no. 1405), the fund fromState Key Laboratory for Strength and Vibration of Mechani-cal Structures (Xi’an Jiaotong University, no. SV2015-KF-02),


the fund from State Key Laboratory of Structural Analysis forIndustrial Equipment (Dalian University of Technology, no.GZ1408), and the fund fromState Key Laboratory ofMechan-ical System and Vibration (Shanghai Jiaotong University, no.MSV201611).

References

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