Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology http://pij.sagepub.com/

Performance analysis of multi-leaf oil lubricated foil bearing Liguo Hu, Guanghui Zhang, Zhansheng Liu, Ruixian Ma, Yu Wang and Jinfeng Zhang Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology published online 27 February 2013 DOI: 10.1177/1350650113475560 The online version of this article can be found at: http://pij.sagepub.com/content/early/2013/02/27/1350650113475560

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Original Article

Performance analysis of multi-leaf oil lubricated foil bearing

Proc IMechE Part J: J Engineering Tribology 0(0) 1–18 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1350650113475560 pij.sagepub.com

Liguo Hu, Guanghui Zhang, Zhansheng Liu, Ruixian Ma, Yu Wang and Jinfeng Zhang

Abstract The traditional bearing applied in the turbo-pump for the hydraulic servo system is rolling element bearing. To satisfy the demand of the high rotating speed for turbo-pump, the oil lubricated foil bearing can be employed in the rotor system. For the working liquid of the servo system is oil and the rotor for the turbo pump is submerged in the hydraulic oil, the bearing has to operate in an oil-rich environment, where the air bearing cannot be employed. The theoretical analysis and numerical simulation are carried out in this study to investigate the static and dynamic characteristics of multi-leaf oil lubricated foil bearing. For the structure form of the multi-leaf foil bearing with five symmetrical arrangements, the foil deformation equation and the Reynolds equation are solved coupled by successive over relaxation method, where the Reynolds boundary condition is employed. Then the load capacity, lift-off speed and static equilibrium position are acquired. By deriving the dynamic deformation equation of the foil, the dynamic stiffness coefficients and damping coefficients are obtained based on the perturbation method. The effect of the rotating speed and perturbation frequency on dynamic characteristics is analyzed. It indicates that the load capacity of the multi-leaf foil bearing is smaller than that of the fixed geometry oil bearing without foil deformation, whereas the stability of the bearing is increased. Keywords Multi-leaf oil lubricated foils bearing, static equilibrium position, damping coefficients, stiffness coefficients Date received: 18 September 2012; accepted: 2 January 2013

Introduction With the increased requirements for the energy density of power machinery, the rotating equipment needs to be working under harsh conditions like high rotating speed, extreme temperature and severe wear. The bearing is a key component of the high rotating speed turbo machinery. In the early 1970s and late 1960s, ASAC (Allied Signal Aerospace Co.) developed a multi-leaf cantilever gas foil bearing under the U.S. Air Force and NASA’s support to satisfy the demand of the stability and longevity for high-speed turbo machinery. Compared with the conventional bearings, the essential characteristics depend on the ﬂexible bearing surface, and the working principles belong to the elastic hydrodynamic lubrication areas. The foil structure is ﬂexible, so the gas foil bearing can establish the ﬁlm thickness according to the speed and load. The gas foil bearing can tolerate the angular deviation and a certain degree of local clearance variation generated from the misalignment of the shaft. The gas foil bearing’s structural damping

can suppress the vibration of the rotor and have excellent stability. Although the development and application of multi-leaf gas foil bearing got the powerful support from the U.S. government, the information on the bearing manufacturing process has not been published.1 In the application process, most of the lubricated medium for the multi-leaf foil bearing is gas. So the load capacity of multi-leaf gas foil bearing is insuﬃcient, which depends on the foil materials and foil manufacturing process. As the viscosity of the oil is much greater than the gas, the oil lubricated multileaf foil bearing can increase load capacity eﬀectively, School of Energy Science and Engineering, Harbin Institute of Technology, People’s Republic of China Corresponding author: Guanghui Zhang, School of Energy Science and Engineering, Harbin Institute of Technology, Main West Street, 92, Harbin 150001, People’s Republic of China. Email: [email protected]

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decrease the demands of the coating on the foils. Compared with the gas foil bearing, although the oil foil bearing has large friction loss and heat generation, it is suitable for the high speed turbo bump of hydraulic servo system, where the lubricated oil is abundant. But compared with the high speed rolling element bearing, the wear and fatigue property can be improved signiﬁcantly. Xu et al.2 introduced the design and experiment of the oil lubricated ﬁve-leaf foil bearing test-bed. The experimental results indicated that the oil lubricated foil bearing can oﬀer high rotating velocity, long life and lower friction loss. Until now, nearly no paper concerned with modeling method of this type oil lubricated foil bearing. This article focuses on the theoretical modeling of the oil lubricated foil bearing, which will make the foundation for the design of this kind bearing for turbo pump of the hydraulic servo system. The key factor of investigating the static and dynamic characteristics for multi-leaf oil foil bearing is to solve the foil deformation equation and the Reynolds equation coupled. This problem belongs to the domain of elastic hydrodynamic lubrication. As the literature about the oil lubricated foil bearing is rare, the study about the gas foil bearing is presented. Oh and Rohde3 assumed a particular shape of the gas ﬁlm, and established the foil deformation equation in the case of considering foil bending eﬀect. The friction between the foils and the friction between the foil and journal surfaces were considered. The foil deformation equation and the gas lubricated Reynolds equation was solved coupled by the ﬁnite element method. The characteristics of multi-leaf gas foil bearing including the lift-oﬀ speed and minimum gas ﬁlm thickness was obtained. Nagaraj4,5 described the geometrical structural characteristics of multi-leaf gas foil bearing in detail. The foil deformation equation and the Reynolds equation was solved and the pressure distribution, ﬁlm thickness, foil deformation was obtained. Reddy et al.6 made further study on the basis of the work of Nagaraj. The foil deformation and contact force between the adjacent foils were calculated by ﬁnite element method. Then the static equilibrium position and dynamic characteristics of gas foil bearing were presented. The ﬁve-leaf oil lubricated foil bearing is studied in this article. The structural geometry of the foil bearing is analyzed and the ﬁlm thickness expression of the clearance is obtained. Based on the work of Nagaraj,4 the foil is assumed as the cantilevered curved beam, and the reaction between the foils is point contact condition. By employing the Cartesian theorem, the deformation equation of elastic curved foil is solved. By using the Christopherson Algorithm, the ﬁlm rupture can be simulated by the Reynolds boundary condition. The Reynolds equation for oil lubricated foil bearing is solved by the ﬁnite diﬀerence scheme and successive over relaxation. By solving the foil

deformation equation and Reynolds equation coupled, the static characteristics of the bearing are got. By deriving the dynamic deformation equation of the foil, the perturbation method can be employed to solve the dynamic stiﬀness and damping coeﬃcients.

Mathematical modeling Governing equation and dimensionless form The following assumptions are employed for incompressible lubrication of journal bearing: the thickness of the oil ﬁlm is relative small compared with the length of the bearing, so the pressure gradient along the ﬁlm thickness is set as zero. The ﬂow in the bearing clearance is laminar ﬂow. The lubrication oil is isotropic, so the viscosity along the ﬁlm thickness is constant. The inertia eﬀect of the oil is ignored. The classical Reynolds equation for incompressible lubrication is as follows 1 @ h3 @p @ h3 @p 1 @h @h þ ¼ þ R2 @ 12 @ @z 12 @z 2 @ @t

ð1Þ

By introducing the dimensionless parameters h¼

h p z 6R2 12R2 , ¼ , p¼ , z¼ , 1 ¼ 2 pa C3 C pa 0:5L pa C2

The incompressible dimensionless oil lubricated Reynolds equation is as follows 2 @ 2R @ @h @h 3 @p 3 @p h h þ ¼ 1 þ 2 @ @ L @z @z @ @t ð2Þ

Differential format for Reynolds equation The oil ﬁlm pressure distribution is symmetric on middle line of the bearing, so the calculation domain is reduced to half of the bearing length, where the symmetry boundary condition is applied. The calculation domain is meshed as shown in Figure 1(a). The nodes i ¼ 1 and i ¼ nx are corresponding to the circumferential coordinates ¼ 0 and ¼ 2. The nodes j ¼ 1 and j ¼ nz are corresponding to the circumferential coordinates z ¼ 0.5L and z ¼ 0. The pressure of any arbitrary nodes (i, j), 1 4 i 4 nx, 1 4 j 4 nzþ1 is concerned with its surrounding nodes. To improve the accuracy, the nodes are inserted in the meshing element on half-step adjacent to node (i, j), which is shown in Figure 1(b). The dimensionless pressure expression on node (i, j) pi, j ¼

1 A6i, j þ 2 h_ A1i, j pi1, j þ A2i, j piþ1, j A5i, j þ A3i, j pi, j1 þ A4i, j pi, jþ1 ð3Þ

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(a)

(b)

Figure 1. (a) Calculation domain; (b) the half step insertion node in mesh element.

where A1i, j ¼

1 3 h 2 i1=2,j

A2i, j ¼

1 3 h 2 iþ1=2, j

A3i, j ¼

1 2R 2 3 hi, j z2 L

condition. This procedure for Reynolds boundary condition is ﬁrst proposed by Christopherson, and proved by Cryer with a strict mathematical method. As shown in Figure 1(a), the number of iterations is the number that updates the pressure in all nodes of the computational domain. So the pressure for k-th iteration is as follows pðkÞ i, j ¼

1 2R 2 ð Þ hi, j : z2 L ¼ A1i, j þ A2i, j þ A3i, j þ A4i, j

A4i, j ¼ A5i, j

A6i, j ¼

1 1 hiþ1=2, j hi1=2, j

The boundary condition and calculation procedure for Reynolds equation For the actual oil lubricated bearing, the oil ﬁlm may have rupture area where the cavitation happens. In this study, the Reynolds boundary condition is employed. In the Reynolds boundary, the oil ﬁlm starts at the maximum of the oil ﬁlm thickness, and terminates at the position where pressure derivation equaling to zero. It means that the oil ﬁlm rupture most likely occurs on the position that the pressure is low. The oil ﬁlm rupture aﬀects the pressure distribution of the bearing, which coincides with the actual situation and is called the free boundary condition. In Figure 1(a), the thick solid line O is the hint of the ﬁlm rupture position that is to meet the Reynolds boundary condition. For the exact location of O is a curve to be determined. This is an undetermined boundary condition problem. Christopherson Algorithm7 is a simple algorithm for solving the problem of the free boundary condition, and the results can satisfy Reynolds boundary

1 ðk1Þ A6i, j A1i, j pðkÞ i1, j þ A2i, j piþ1, j A5i, j ðk1Þ þ A4 p þ A3i, j pðkÞ i, j i, j1 i, jþ1

where, k ¼ 1,2 . . . is the number of iteration. Updating ðkÞ ðk1Þ pðkÞ , the expression is as follows i, j with pi, j and pi, j ðkÞ ðk1Þ pðkÞ ¼ p p þ pðk1Þ i, j i, j i, j i, j

ð4Þ

where is the relaxation coeﬃcient. For the equations set whose convergence properties is better, >1 is taken to reduce the number of iterations and accelerate convergence, which is called successive over relaxation (SOR). The SOR is employed to accelerate convergence in this article. When the kth iteration is ﬁnished, the dimensionless pressure is set to 1 for the nodes whose dimensionless pressure is smaller than 1. The following equation is considered ðkÞ pðkÞ i, j ¼ maxð1:0, pi, j Þ

ð5Þ

The convergence criterion is that the diﬀerences between two adjacent iterative calculations satisfy equation (6) Pnx Pnz ðkÞ ðk1Þ i¼1 j¼1 pi, j pi, j 5 Pnx Pnz ðk1Þ i¼1 j¼1 pi, j is usually smaller than 106.

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ð6Þ

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y Inscribed Circle

Roling Circle

y1

o1 o2

Ψ

o

x

o4

Ω

Foil

o5

Journal

In Figure 3(a), according to the geometrical relationship, the abduction angle can be obtained8 ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i 2 2 2 2 Rf 0:5Ro 0:25Rb Rb Ro ¼ arcsin Rf Rf Ro

Ro 2 þ arccos 1 sin 2 1 5 2 Rf ð7Þ

Bearing housing

x1

o3

The installation angle of the foil, the abduction angle T of tangent point Tp, the abduction angle C of the contact point Cp, the expressions are as follows

θ

Figure 2. The structure of five-leaf foil bearing.

The pressure distribution is obtained when the convergence criterion is satisﬁed. By employing the above point-by-point SOR iteration, the calculated pressure distribution will meet the Reynolds boundary condition.

¼ arccos

2 R2f þ R2b Rf Ro

ð8Þ

2Rf Rb

2 R2f þ Rf Ro R2b T ¼ arccos 2Rf Rf R0

ð9Þ

C ¼ T ð T Þ

ð10Þ

The foil center O1, contact point Cp and the coordinate of point p with any abduction angle of the foil are as follows

The geometry and computational domain of multi-leaf foil bearing

The geometry of multi-leaf foil bearing. The ﬁve-leaf foil bearing which is arranged symmetrically on the y-axis is analyzed in this article. Five cantilevered foils are overlapped counterclockwise as shown in Figure 2. The ﬁxed end for the foil is installed on the housing of the bearing, and the free end laps on the next foil. The inscribed circle for the foils group is tangential with each foil and a certain clearance is kept. O1, O2, . . . ,O5 are the centers of the ﬁve separated foils, which formed the rolling circle. As the geometry parameters and installation angle are same for each foil, only the foil A is studied to facilitate the analysis. Foil A is taken out from the bearing and set coordinate system x1Oy1 as the reference system, which is shown in Figure 3. The radius angle for the foil is 2p/5. In Figure 3(a), the unknown parameters in coordinate x1Oy1 are as follows: the abduction angle D of the foil, the abduction angle C of the foil contact point Cp, the abduction angle T of the foil inscribed circle’s tangent point Tp, the installation angle of the foil, the coordinate (x1O1, y1O1) of the foil center O1, the coordinate (x1p, x1p) of any point on the foil. In Figure 3(b), the unknown parameters: the coordinate (xO1, yO1) of the foil center O1, the coordinate (xO0 , yO0 ) of axial center O0 , the coordinate (xp, yp) of any point p on the foil, the ﬁlm thickness h0 on any point p on the foil.

x1O1 y1O1 x1Cp y1Cp x1p y1p

¼

¼

¼

Rf sinð Þ Rf cosð Þ Rb x1O1 y1O1

x1O1 y1O1

þ

þ

ð11Þ

Rf sinðC þ Þ Rf cosðC þ Þ

Rf sinðC þ Þ Rf cosðC þ Þ

ð12Þ

ð13Þ

In Figure 3(b), nominal radius clearance Cn ¼ R0R, the transition angle between the coordinate system xOy and x1Oy1 ¼ arctan x1Cp =y1Cp

ð14Þ

Set ¼ , then the foil center O1 in the coordinate system xOy, axial center O0 , the coordinate of any point p on the foil are as follows

xO1 yO 1

xO 0 yO 0 xp yp

x ¼ ½TðÞ 1O1 y1O1

¼

en sinðnÞ en cosðnÞ

x ¼ ½TðÞ 1p y1p

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ð15Þ

ð16Þ

ð17Þ

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(a)

(b)

Figure 3. The schematic of geometrical depiction for foil A.

where Tð Þ ¼

cos

sin

sin

cos

* * In Figure 3(b), vector a and vector b are expressed as * * a ¼ xO1 xO0 ,yO1 yO0 b ¼ xp xO0 ,yp yO0 As a foil can be treated as a part of rigid self-acting bearing, the expression of the ﬁlm thickness h0 on any point p of the foil is * h0 ¼ Rf R þ a cos ¼ Rf R * * * a b þ a * * a b

ð18Þ

Combined with the rest geometrical parameters of the foil, the foil center Oi and the coordinates of any point on the foil pi are presented in equations (19) and (20), where i ¼ 1,2,3,4,5

xOi yO i xp i yp i

x ¼ ½Tðði 1ÞÞ O1 yO 1

¼ ½Tðði 1ÞÞ

xp yp

Figure 4. The calculation domain for five-leaf foil bearing.

Division of the calculation domain The calculation domain for the ﬁve leaves foil bearing is presented in Figure 4. As the oil ﬁlm thickness is very small which can be ignored compared with the foil size, the computational domain can be expanded on the foil in the direction of . Every foil is divided with mesh size of nxnz. The area for a mesh element is Se. The foils are connected by contact points. The point on the end side of the ith foil is coincided with the point on the start side of the iþ1-th foil. The boundary condition for the whole calculation domain is as follows: For the symmetrical boundary condition

ð19Þ pð1 : nx þ 1, 1Þ ¼ p0 pð1 : nx þ 1, nz þ 1Þ ¼ pð1 : nx þ 1, nz 1Þ

ð20Þ

For the periodic boundary condition The ﬁlm thickness of any point for the foil bearing * * * ai bi h0i ¼ Rf R þ a i * * a i bi

pð0, 1 : nz þ 1Þ ¼ pðnx, 1 : nz þ 1Þ pðnx þ 1, 1 : nz þ 1Þ ¼ pð1, 1 : nz þ 1Þ

ð21Þ Also the Reynolds boundary condition will be satisﬁed by the Christopherson Algorithm.

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Figure 6. The schematic of foil forces analysis.

The generalized virtual force F 0 is set as zero, so the radial displacement j for node j under the radial force F can be expressed as follows Figure 5. The schematic of foil deformation analysis.

Z

Z 0d þ

j ¼

0

The solving of the foil deformed flexibility matrix and reaction force The pressure in the clearance of the multi-leaf foil bearing can generate the bearing load capacity, and it will make the foil deform. The clearance will be changed and aﬀect the pressure distribution. The deformation of the foil with oil ﬁlm pressure and interaction force will make the foundation for the analysis of the bearing. In this article, the foil is assumed as the cantilevered curved beam and the ﬂexibility matrix for the curved beam will be obtained. By analyzing the forces for the foil bearing, the reaction forces between foils can be solved. The flexibility matrix of the foil deformation. The foil can be simpliﬁed as the cantilevered curved beam in Figure 5. The width of the beam is shown in Figure 4, where the area is formed by the dotted line. The radial force F acting on the node i with abduction angle of will result in the radial displacement j for the node j with arbitrary abduction angle of . With the assumption of small deformation, the Castigliano’s theorem can be employed in this article. For the case of 4, the radial force and the generalized virtual force F 0 for j node with abduction angle of act on the foil simultaneously Mð Þ ¼ F Rf sinð Þ Mð Þ ¼ F Rf sinð Þ þF 0 Rf sinð ð ÞÞ

2 ½0, Þ For the curve i j 2 ½ , For the curve j A

Radial deformation j is Z

j ¼ 0

@Mð Þ Mð Þ Rf d @F0 EI

¼

F R3f 4EI

F Rf sinð Þ Rf sinð ð ÞÞ Rf d EI

½2 cosð Þ þ sinð Þ

sinð þ Þ 2 ½0,

Similarly, for the case of 44, the radial displacement is expressed

j ¼

F R3f

½2 cosð Þ þ sinð Þ 4EI sinð þ Þ 2 ½,

The ﬂexibility matrix U(i, j) indicates the displacement of node j with unit force acting on the node i for the cantilevered beam. Then the ﬂexibility matrix is as follows

Uði, jÞ ¼

8 > R3f > > ½2 cosð Þ þ sinð Þ > > > 4EI > > > > < sinð þ Þ 14j4i > > R3f > > > ½2 cosð Þ þ sinð Þ > > 4EI > > > : sinð þ Þ i4j4n

Reaction force between the foils. The foils are contacted with each other at contact point, as shown in Figure 6. The radial displacements of two foils at the contact point are the same. By ignoring the friction force, the interaction force is only radial reaction force. The dynamic oil ﬁlm force acting on the jth foil is Pnj ¼ pjSe. The reaction force on the jth foil generated from the j1 foil is Ni. The reaction force on the jth foil generated from the jþ1 foil is Niþ1. The forces mentioned above are balanced on the j the foil.

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Combing the ﬂexibility matrix, the following equation can be derived Niþ1 Uðn, 1Þ þ Pnj ðkÞUðk, 1Þ þ Ni Uð1, 1Þ ¼ Ni Uðn, nÞ þ Pnj1 ðkÞUðk, nÞ þ Ni1 Uð1, nÞ where k ¼ 1, 2, . . . , n. Further Ni ½Uð1, 1Þ þ Uðn, nÞ Niþ1 Uðn, 1Þ Ni1 Uð1, nÞ ¼ Pnj1 ðkÞUðk, nÞ Pnj ðkÞUðk, 1Þ ð22Þ For ﬁve-leaf foil bearing, it can be written in matrix form ½A½N ¼ ½Pn½Uc

Figure 7. The schematic of the static equilibrium position for the shaft in foil bearing.

ð23Þ

where ½N ¼ ðN1 , N2 , N3 , N4 , N5 ÞT

ð24Þ

0

Uð1, 1Þ þ Uðn, nÞ Uðn, 1Þ 0 B Uð1, nÞ Uð1, 1Þ þ Uðn, nÞ Uðn, 1Þ B B B ½A ¼ B 0 Uð1, nÞ Uð1, 1Þ þ Uðn, nÞ B 0 0 Uð1, nÞ @ Uðn, 1Þ 0 0 pnz 1ð1Þ pnz 1ðnÞ pnz 5ð1Þ B B pnz 2ð1Þ pnz 2ðnÞ pnz 1ð1Þ B ½Pn ¼ Se B B pnz 3ð1Þ pnz 3ðnÞ pnz 2ð1Þ B @ pnz 4ð1Þ pnz 4ðnÞ pnz 3ð1Þ pnz 5ð1Þ pnz 5ðnÞ pnz 4ð1Þ T ½Uc ¼ U1, 1 , . . . , Un, 1 , U1, n , . . . , Un, n

0

The support reaction force array [N] can be obtained by solving equation (23), where the interaction forces between foils are calculated. Expression of the film thickness. From equation (18), the ﬁlm thickness of the foil bearing with foil deformation is h ¼ h0 þ hf

ð25Þ

where hf is the radial deformation from the ﬁlm pressure and the reaction force on the foil, namely, the deformation of the point i on the jth foil hf jðiÞ ¼ Nj Uð1, iÞ Njþ1 Uðn, iÞ n X Pnj ðkÞ Uðk, iÞ þ k¼1

pnz 5ðnÞ

0 0

Uð1, nÞ 0

Uðn, 1Þ Uð1, 1Þ þ Uðn, nÞ

0 Uðn, 1Þ

Uð1, nÞ

Uð1, 1Þ þ Uðn, nÞ

1

1 C C C C C C A

C pnz 1ðnÞ C C pnz 2ðnÞ C C C pnz 3ðnÞ A pnz 4ðnÞ 5ð2nÞ

where i ¼ 1, 2 . . . ,n, j ¼ 1, 2 . . . 5. Njþ1 ¼ N1 when j ¼ 5.

The load capacity and static equilibrium position for the foil bearing As the structure of the ﬁve-leaf foil bearing is not symmetrical, the pressure distribution for the bearing depends on the eccentricity e and the attitude angle ’. The variables mentioned above are demanded to obtain the static equilibrium position simultaneously. As shown in Figure 7, the following relationship needs to be satisﬁed Fy ¼ W

ð26Þ

Fx ¼ 0

ð27Þ

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The angle 0 between the wire (xj,yj)-O0 and Fy is deﬁned as follows

H 0 ¼ h0 þ hf

ð34Þ

0 ¼ arctan yi yO0 , xi xO0

Hx ¼ hx þ ivhx_

ð35Þ

Then

Hy ¼ hy þ ivhy_

ð36Þ

Z2 Z0 p cos R dzd

Fy ¼ 2

ð28Þ

0 L=2

Z2 Z0 Fx ¼ 2

p sin R dzd

ð29Þ

H0 is the ﬁlm thickness for steady-state operation. H is the perturbed ﬁlm thickness. The perturbed ﬁlm pressure P can be expanded by the Taylor equation about the steady-state operated pressure P0 * * * * p ¼ P0 þ px x þ py y þ px_ x_ þ py_ y_

0 L=2

The pressures p (i, j) on each node have been obtained in the section ‘The boundary condition and calculation procedure for Reynolds equation’.

Dynamic characteristics of the multi-leaf foil oil lubricated bearings In the study of the dynamic coeﬃcients for the multileaf foil bearing, the elastic deformation of the foil and the ﬁlm pressure are coupled. Thus, the perturbation of the pressure distribution will cause the perturbation deformation of the foil, and then aﬀect the perturbation of the ﬁlm pressure. So the perturbation method will be employed in this study to derive the expression of the perturbed foil deformation. By coupling with the perturbed Reynolds equation, the stiﬀness and damping coeﬃcients can be obtained. The perturbation of the foil deformation. If the journal perturbs around the static equilibrium position (xO0 , yO1) of the axial center O0 when the perturbation frequency is u rad/s, the position of the axial center can be expressed as

¼ P0 þ P ¼ P0 þ Px xeivt þ Py yeivt

ð37Þ

where Px ¼ px þ ivpx_

ð38Þ

Py ¼ py þ ivpy_

ð39Þ

Taking equation (21) into equation (25), the following equation can be obtained * * H ¼ h0 þ hf ¼ x sin y cos * * þ Hf x x_ þ Hf y y_ ¼ sin þ Hf x xeivt þ cos þ Hf y yeivt ð40Þ where Hf x ¼ hf x þ ivhf x_

ð41Þ

Hf y ¼ hf y þ ivhf y_

ð42Þ

* x ¼ xO0 þ x ¼ xO0 þ xeivt

ð30Þ

Comparing equation (40) with equation (33)

* þ y ¼ yO0 þ yeivt

ð31Þ

Hx ¼ sin þ Hf x

ð43Þ

Hy ¼ cos þ Hf y

ð44Þ

y ¼ yO 0

* * where x and y are the vectors of the perturbation displacement; x and y are the amplitudes of the vectors * * x_ ¼ ivxeivt ¼ iv x * * y_ ¼ ivyeivt ¼ iv y

ð32Þ

Combined with equation (25), the Taylor expansion of the perturbed ﬁlm thick function h for steady-state operation can be obtained * * * * h ¼ h0 þ hf0 þ hx x þ hy y þ hx_ x_ þ hy_ y_ ¼ H0 þ H ¼ H0 þ Hx xeivt þ Hy yeivt

* * where is the angle between vector ai and bi as shown in Figure 3. As the dotted line range of the calculation domain in Figure 4 has been assumed as the bending beam, the pressure and foil deformation on the nodes can be expressed with the matrix form as follows

T ½pz ¼ p1, nz , p2, nz , . . . , pi, nz , . . . , pnxþ1, nz

ð45Þ

T hf ¼ hf 1, hf 2, . . . , hf i, . . . , hf nx þ 1

ð46Þ

ð33Þ

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By employing equation (23)

along the z-axis is ignored, equation (54) can be simpliﬁed as follows

½N ¼ ½A1 ½Pn½Uc ¼ ½B5ðnxþ1Þ ½pz

ð47Þ

The matrix [B] can be obtained by merging Pnz in equation (23). Substitute equation (47) into equation (25), then hf can be expressed by matrix [Pz] hf ¼ ½UHEðnxþ1Þðnxþ1Þ ½pz

ð48Þ

If the dynamic deformation coeﬃcients matrix [UHE] can be obtained, the [hf] can be got as the matrix [pZ] is known. hf ¼ hf0 þ hf * * ¼ hf0 þ Hfx x þ Hfy y * ¼ ½UHE½Pz0 þ ½UHE½Pzx x * þ ½UHE Pzy y

ð49Þ

Hfx ¼ ½UHE½Pzx

ð50Þ

Hfy ¼ ½UHE Pzy

ð51Þ

The matrix [Pzx], [Pzy] are the partial derivative of * * pressure tox , y . Substituting equations (50) and (51) into equation (40), then the column array of the perturbation deformation for each node can be obtained

½H ¼ sin½ þ Hf x xeivt

þ cos½ þ Hf y yeivt

ð52Þ

The discretization and solution of the perturbed Reynolds equation. Substituting equations (33) and (37) into the Reynolds equation (equation (1)), the following equation can be got by ignoring the second or higher order minute amount.

@ @P0 @ @P0 @h0 H30 H30 þ R2f ¼ 6R2f @ @z @ @z @

@H @H þ 12R2f @ @t

ð55Þ

@H0 @ @2 @2 þ H30 2 þ R2f H30 2 @ @ @ @z

@P0 @H0 @2 P0 þ 3H20 2 @ @ @ 2 @ P0 þ 3R2f H20 2 iv 12R2f @z

ð56Þ

G2 ¼ 6H0

@P0 6R2f @

ð57Þ ð58Þ

Substitute equations (37) and (40) of P, H into the above equations, the following can be obtained G1 Px þ G2 Hf x þ G3

G1 Py þ G2 Hf y þ G3

@Hf x ¼ G2 sin G3 cos @ ð59Þ @Hf y ¼ G2 cos G3 sin @ ð60Þ

Substitute equation (50) into equation (59), then the perturbed Reynolds equation for any node ði, jÞ on the calculation domain is as follows G1 Px ði,jÞ þ TEMðiÞ ½Pz x ¼ G2 sin ðiÞ G3 cos ðiÞ ð61Þ where TEMðiÞ ¼ G2 UHEði, 1 : nx þ 1Þ 1 G3 ½UHEði þ 1, 1 : nx þ 1Þ 2

ð53Þ UHEði 1, 1 : nx þ 1Þ

@ 3 @P 2 @P0 H0 þ 3H0 H @ @ @

2 @ 3 @P 2 @P0 þ Rf H0 þ 3H0 H @z @z @z ¼ 6R2f

G1 ¼ 3H20

þ

@H ¼0 @

where

G3 ¼ 3H20

Then

G1 P þ G2 H þ G3

ð62Þ

The ﬁrst right hand term of equation (62) can be discrete by the ﬁnite diﬀerence method as follows G1 Px ði, jÞ ¼ A1 ði, jÞ Px ði 1, jÞ þ A2 ði, jÞ Px ði, jÞ ð54Þ

Equation (53) is the steady-state operation Reynolds equation. Equation (54) is the perturbed Reynolds equation. As the ﬁlm thickness variation

þ A3 ði, jÞ Px ði þ 1, jÞ þ A4 ði, jÞ Px ði, j 1Þ þ A5 ði, jÞ Px ði, j þ 1Þ ¼ A2 ði, jÞ Px ði, jÞ þ TXði, jÞ

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ð63Þ

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where

For the oil ﬁlm rupture area

1 H3 ði, jÞ 2 0 3 H2 ði, jÞ ½H0 ði þ 1, jÞ H0 ði 1, jÞ 4 2 0 A2 ði, jÞ ¼ A1 ði, jÞ A3 ði, jÞ A4 ði, jÞ A5 ði, jÞ 1 3 H3 ði, jÞ þ H2 ði, jÞ ½H0 ði þ 1, jÞ A3 ði, jÞ ¼ 2 0 4 2 0 H0 ði 1, jÞ 2 Rf A4 ði, jÞ ¼ A5 ði, jÞ ¼ H30 ði, jÞ z

A1 ði, jÞ ¼

For the case of j 6¼ nz, substituting equation (63) into equation (61)

Px ði, jÞ ¼ Py ði, jÞ ¼ 0 To solve Px(i,j) and Py(i,j), SOR method is used in this article. Dynamic stiffness and damping coefficients. As shown in equations (38) and (39), the complex variable Px(i,j) and Py(i,j) can be obtained, where the real and imaginary part are respectively the derivative of the ﬁlm pressure to the journal displacement and velocity Px ði, jÞ ¼ px ði, jÞ þ iv px_ ði, jÞ Py ði, jÞ ¼ py ði, jÞ þ iv py_ ði, jÞ Then

TXði, jÞ þ TEMðiÞ ½Pzx þG2 ði, jÞ sin ðiÞ þ G3 ði, jÞ cos ðiÞ Px ði, jÞ ¼ A2 ði, jÞ

ð64Þ

For the case of j ¼ nz, the coeﬃcient of Px(i,nz) is deﬁned as E(xi) EðiÞ ¼ A2 ði, nzÞ þ G2 ði, nzÞ UHEði, iÞ 1 G3 ði, nzÞ ½UHEði þ 1, iÞ UHEði 1, iÞ þ 2 ¼ A2ði, nzÞ þ EFðiÞ TXði, jÞ þ TEMðiÞ ½Pzx þ G2 ði, jÞ sin ðiÞ Px ði, jÞ ¼ EðiÞ þG3 ði, jÞ cos ðiÞ EFðiÞ Pzx ði nzÞ EðiÞ

The Py(i,j) on the point (i,j) can also be expressed as the equation mentioned above. The solving procedure for Px(i,j) and Py(i,j) needs to satisfy the cyclic boundary condition as follows, which is shown in Figure 4 Px ð0, 1 : nzÞ ¼ Px ðnx, 1 : nzÞ

KDxx

Px

¼ 2 0

Py

0

sin Rf ddz cos

L=2

Z2 Z0 ¼ 2

KDyx

Z2 Z0

sin cos

Rf ddz

L=2

Where the matrix [KD] is as follows ½KD ¼

Kxx þ iv Dxx

Kxy þ iv Dxy

Kyx þ iv Dyx

Kyy þ iv Dyy

C0 Re½KD K ¼ W C0 Im½KD D ¼ Wv The matrix K and D are dimensionless form of the stiﬀness and damping matrix.

Numerical simulations Calculation of the foil deformation

Py ð0, 1 : nzÞ ¼ Py ðnx, 1 : nzÞ Px ðnx þ 1, 1 : nzÞ ¼ Px ð1, 1 : nzÞ Py ðnx þ 1, 1 : nzÞ ¼ Py ð1, 1 : nzÞ Symmetric boundary conditions Px ð1 : nx, nz þ 1Þ ¼ Px ð1 : nx, nz 1Þ Py ð1 : nx, nz þ 1Þ ¼ Py ð1 : nx, nz 1Þ Fixed value boundary conditions Px ð0 : nx þ 1, 1Þ ¼ 0

KDxx KDyx

Px ð0 : nx þ 1, 1Þ ¼ 0

To verify the accuracy of the foil deformation calculation with the Castigliano’s theorem, the comparison between the Castigliano’s theorem and the ﬁnite element method is presented. The ﬁnite element method is carried out by the software Abaqus. The simulation parameters are as follows: ¼ 80 , ¼ 60 , Rf ¼ 10 mm, the width of the curved beam is 1 mm, thickness of the beam section is 0.5 mm. The curved beam is divided into eight nodes. The results are presented in Table 1. It indicates that the results are fundamental consistence. It means that the Castigliano’s theorem can be employed to solve the ﬂexibility matrix U(i, j) accurately.

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Table 1. The result of the radial displacement for curved beam. Radial deformation (mm) Node numbering

Abduction angle ( )

Castigliano theorem results

Finite element method results

1 2 3 4 5 6 7 8

10 20 30 40 50 60 70 80

0.00579737 0.0220320 0.0465028 0.0764881 0.108888 0.140385 0.168021 0.190551

0.00582508 0.0220927 0.0465999 0.0766226 0.109059 0.140588 0.168201 0.190704

Table 2. The parameters of five multi-leaf foil bearing. Item

Description

Value

NF Rf TI E R Rb Ro L W P0 nx nz

Number of the foil Radius of the foil (m) Thickness of the foil (m) Elastic modulus of the foil (Pa) Radius of the journal (m) Radius of the bearing housing (m) Radius of the tangent circle (m) Length of the bearing (m) Bearing load capacity (N) Viscosity of the lubricated oil (Pa s) Ambient pressure (MPa) The number of the element for the foil along circumferential direction The number of the element for the foil along axial direction

5 0.01 0.0005 2.11011 0.007 0.008 0.00705 0.01 1.5 0.59610-3 0.010325 51 21

The static characteristics of oil-lubricated foil bearing The foil structure is shown in Figure 1, and the detail parameters are presented in Table 2. The static characteristics of the ﬁve-leaf oil foil bearing are simulated. The comparisons between the cases with and without foil deformation are presented. With the parameters in Table 2, the nominal eccentricity is 0.02103 mm, and the nominal attitude angle is 20 . The rotating speed is 10,000 r/min. As shown in Figure 2, the ﬁve leaves are completely identical. In Figure 8(a), the three dimensionless pressure distribution of the foil bearing are presented where foil deformation is considered. In Figure 8(b), when the foil deformation is ignored, the center of the journal O0 is close to the foil A and E. The minimum ﬁlm thickness locates between the foil A and the journal, which is equaling to 0.62 C0 in Figure 9. The maximum pressure is on the foil A, and it is 5.9 P0. When the foil deformation is considered, the minimum ﬁlm thickness is still near the foil A, but increased to 0.76

C0 that is presented in Figure 10. Similar to the case of ignoring the foil deformation, the maximum pressure is still on the foil A, but decreased to 4.8 P0. The pressure of the bearing with the foil deformation considered is higher than that of ignoring the foil deformation in general. As the maximum pressure happens on the foil A and foil E, the maximum deformation of the foil is on the foil A, which is shown in Figure 10. The maximum foil deformation of the bearing is 0.14 C0, which is shown in Figure 11. For the cases with and without the foil deformation, the maximum ﬁlm thickness happens on the overlap joint of adjacent foils. From the numerical simulation, it indicates that for the area of maximum ﬁlm thickness, the pressure of the ﬁlm is a little higher than that of the atmosphere pressure. So the rupture area of the oil ﬁlm is relative small. When the foil deformation is considered, the domain of higher oil ﬁlm pressure increases on foil A and foil E. The oil ﬁlm rupture area on foil E reduces obviously. It means that the foil deformation can enlarge the pressure area of the bearing, and reduce the area of

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Figure 8. The pressure distribution for oil lubricated foil bearing: (a) pressure distribution for the foil bearing with foil deformation; (b) the comparison of the pressure distribution for foil’s middle line with the case of considering and ignoring the deformation.

Figure 9. The film thickness distribution ignoring the deformation of the foil.

Figure 10. The film thickness distribution considering the deformation of the foil.

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ﬁlm rupture. So the stability of the bearing is improved when the foil deformation happens. As the maximum pressure decreases, so the load capacity of the bearing with foil deformation is smaller than that of without foil deformation. For the structure model in Table 2, the lift oﬀ speed for the bearing is studied in this article. By numerical simulation, when the rotating speed is smaller than 970 r/min, the bearing load capacity is smaller than 1.5 N. So the rotating speed of 970 r/min is the case that the shaft and the foil has been separated by the lubricated oil, which is called the lift oﬀ speed. As the rotating speed increases to 100,000 r/min, the static equilibrium position locus of the shaft center is depicted in Figure 12. The variation of the nominal attitude angle with the nominal eccentricity is presented in Figure 13. In Figure 12, as the rotating speed increases, the center of the shaft is gradually approaching to the center of the bearing, where the nominal eccentric ratio decreases. But varied range of the nominal attitude angle is about 6 . The locus of the shaft center moving to the bearing center is nearly the straight line. This phenomenon is generated from the symmetrical arrangement and elastic deformation of the foils, which leads to the motion of the shaft along the load direction. The displacement of the shaft perpendicular to the load direction is relative small. So the force of direction ’n is small, where the destabilizing force of the oil ﬁlm is small. It indicates that the ﬁve-leaf foil bearing has an excellent stability. In Figure 14 the variation of the nominal Sommerfeld coeﬃcient with the nominal eccentricity is presented, where the Sommerfeld coeﬃcient is deﬁned as follows Sn ¼ RLðR=C0 Þ2 =W

O

90

0.0 0.2

75

0.4 60

0.6 0.8

45

1.0

30 0

15

Nominal attitude angle ϕ n

Figure 12. The orbit of static equilibrium position for five multi-leaf foil bearing.

Figure 13. The relation between nominal attitude angle and nominal eccentricity with the balancing of shaft center.

Figure 11. The film thickness distribution considering the deformation of the foil.

Figure 14. The relation between the nominal Sommerfeld coefficients and nominal eccentricity.

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Figure 15. The stiffness coefficients varying with the nominal Sommerfeld coefficient.

As the eccentricity increases, the Sommerfeld coefﬁcient decreases. When the eccentricity is small, the variation of the Sommerfeld is severe. It indicates that the shaft center position is stable when the rotating speed is bigger. When the rotating speed is bigger than 10,000 r/min, the eccentricity ratio is 0.2, and the Sommerfeld coeﬃcient is larger than 20. It indicates that the designed ﬁve-leaf foil bearing is suitable for the high rotating speed operation. When the perturbation frequency ratio p ¼ / is equaling to 1 and the speed ranges from 971 to 10,000 r/min, the variation of the stiﬀness coeﬃcients with the nominal Sommerfeld coeﬃcient is analyzed in Figure 15. For the heavy load case, the main stiﬀness on load direction Kyy is slightly bigger than that of Kxx . For the case of high rotating speed and light load capacity, the main stiﬀness of the two direction increases simultaneously as the rotating speed increases. The diﬀerence between the Kyy and Kxx decreases gradually. But Kyy is consistently bigger than Kxx . The cross coupling stiﬀness Kxy is negative when the Sommerfeld coeﬃcient is smaller than 0.5. It varied dramatically when the Sommerfeld coeﬃcient is 0.5. As the Sommerfeld coeﬃcient increases, the Kxy changes to positive value. The cross coupling stiﬀness Kyx is always negative. In Figure 16, the dimensionless damping coeﬃcients varying with the nominal Sommerfeld coeﬃcient are presented, where the perturbation frequency is 1. When the Sommerfeld coeﬃcient is small, the diﬀerence between the two main damping coeﬃcients is obvious. The main damping coeﬃcient Dyy is bigger than the coeﬃcient Dxx . But as the Sommerfeld coeﬃcient increases, the two main damping coeﬃcients increase and the diﬀerence between

them disappears. When the rotating speed is lower than 15,000 r/min, the cross coupling damping coeﬃcients are all negative. They ﬂuctuate as the Sommerfeld coeﬃcient increases, and the tendency is similar. When the Sommerfeld coeﬃcient is in the domain between 2.5 and 3.6, the damping coeﬃcients Dxy and Dyx are changed into the positive value. The coeﬃcient Dyx increases as the Sommerfeld coeﬃcient increases, and the coeﬃcient Dxy becomes negative the absolute value approaches the coeﬃcient Dxy . The variation of the dimensionless stiﬀness and damping coeﬃcients with the perturbation frequency for rotating speed of 3000 r/min, 15,000 r/min and 50,000 r/min are presented in Figures 17 to 20. As shown in Figure 17, the main stiﬀness coeﬃcient Kyy is bigger than Kxx all along. But the diﬀerence between them is neglect able when the rotating speed approaches 50,000 r/min. As the rotating speed is lower at about 3000 r/min, the main stiﬀness coeﬃcients are not insensitive to the variation of the perturbation frequency. When the rotating speed increases to 15,000 r/min, the main stiﬀness coeﬃcients increase as the perturbation frequency increases. This phenomenon can be observed obviously for the case of 50,000 r/min. It can be concluded that the perturbation frequency aﬀects the main stiﬀness coeﬃcients. As the perturbation increases, the main stiﬀness coeﬃcients increase. It indicates that the perturbation frequency ratio has little eﬀect on the cross coupling stiﬀness, which is shown in Figure 18. Even the rotating speed reaches about 50,000 r/min, the variation range for dimensionless cross coupling stiﬀness coeﬃcient is still small. As the perturbation frequency ratio increases, the cross coupling stiﬀness coeﬃcient Kxy decreases,

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Dimensionless damping coefficients

Hu et al.

1

0.1

0.01

D xx D yy D xy positive D xy negative D yx positive

1E-3

D yx negative 10 Nominal Sommerfeld coefficient S n =

μ Ω RL ⎛ R ⎞ ⎜ ⎟ π W ⎝ C0 ⎠

100

2

Dimensionless main stiffness coefficients

Figure 16. The damping coefficients varying with the nominal Sommerfeld coefficient.

40

K yy 3 krpm K xx 3 krpm

35

K yy 15 krpm K xx 15 krpm

30

K yy 50 krpm K xx 50 krpm

50 krpm

25 20 15

15 krpm

10 3 krpm

5 0 0

2

4 6 Perturbation frequency ratio ν / Ω

8

10

Figure 17. The main stiffness coefficients varying with perturbation frequency.

but the coeﬃcient Kyx increases. The dimensionless damping coeﬃcients varying with the perturbation frequency are presented in Figures 19 and 20. The inﬂuence of the frequency perturbation ratio on the dimensionless damping coeﬃcients is apparent when the rotating speed is high. When the frequency perturbation ratio increases, the main damping coeﬃcients Dxx and Dyy decrease. The cross coupling stiﬀness coeﬃcient Dyx decreases and the Dxy increases. The above tendency is not obvious when the rotating speed is low. The above simulation indicates that the perturbation frequency has little inﬂuence on the stiﬀness

and damping coeﬃcients when the rotating speed is low. As the rotating speed is high, the main stiﬀness and damping coeﬃcients changes obviously with the perturbation frequency. As the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcients decreases. The cross-coupling stiﬀness and damping coeﬃcients change little with the perturbation frequency. This phenomenon is similar to the experimental results of gas bump foil bearing in Heshmat and Ku.9 In their study, as the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcient decreases.

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2.5

K xy 3 krpm K yx 3 krpm K xy 15 krpm K yx 15 krpm

50 krpm

2.0 1.5

15 krpm

Dimensionless coupling stiffness coefficients

1.0

K xy 50 krpm K yx 50 krpm

0.5 3 krpm

0.0

3 krpm

–0.5 –1.0

15 krpm

–1.5 –2.0

50 krpm

–2.5 0

2

4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 18. The cross coupling stiffness coefficients varying with perturbation frequency.

50 krpm

5.5

D yy 3 krpm D xx 3 krpm

5.0

D yy 15 krpm D xx 15 krpm

4.5 Dimensionless main damping coefficients

4.0

D yy 50 krpm

3.5

D xx 50 krpm

3.0 15 krpm

2.5 2.0 1.5 1.0 0.5

3 krpm

0

2 4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 19. The main damping coefficients varying with perturbation frequency.

0.15 50 krpm

D xy 3krpm D yx 3krpm

0.10

D xy 15 krpm D yx 15 krpm

15 krpm

0.05

Dimensionless coupling damping coefficients

D xy 50 krpm D yx 50 krpm

15 krpm

0.00

3 krpm

–0.05 –0.10 50 krpm

–0.15 0

2

4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 20. The cross coupling damping coefficients varying with perturbation frequency.

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Conclusions The geometrical structure of the multi-leaf foil bearing with ﬁve symmetrical arranged foils are analyzed in this article. By considering the deformation of the foil, the steady-state incompressible Reynolds equation is solved. The static characteristics such as the lift-oﬀ speed and static equilibrium position are obtained, which make the foundation for the dynamic characteristics of the oil foil bearing. The relation between the perturbed pressure and foil deformation is derived by the perturbation method and the dynamic deformation equation is obtained. By employing the SOR method and combining the boundary condition, the perturbed Reynolds equation is solved. By integrating the pressure, the dynamic parameters including the stiﬀness and damping coefﬁcients concerned with the perturbation is got. The variation of the stiﬀness and damping coeﬃcients with the Sommerfeld parameter is analyzed. The inﬂuence of the perturbation frequency on the dynamic coeﬃcients is presented. As the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcient decreases. This phenomenon is obvious for the case of the high rotating speed. For the cross coupling stiﬀness and damping coeﬃcients, the perturbation frequency has little inﬂuence. The study makes the foundation for the design and analysis of the oil foil bearing for turbo pump, which is employed in the hydraulic servo system. The experiment needs to be carried out in further study to verify the accuracy of the modeling method for oil lubricated foil bearing.

Funding This study was supported by the National Natural Science Foundation of China (grant no. 51106035) and China Postdoctoral Science Foundation funded project (grant no. 2012M510088).

References 1. DellaCorte C, Radil KC, Bruckner RJ, et al. Design, fabrication, and performance of open source generation I and II compliant hydrodynamic gas foil bearings. Tribol Trans 2008; 51(3): 254–264. 2. Xu HJ, Liu ZS, Zhang GH, et al. Design and experiment of oil lubricated five-leaf foil bearing test-bed. J Eng Gas Turb Power 2009; 131: 054505. 3. Oh KP and Rohde SM. A theoretical investigation of the multileaf journal bearing. J Appl Mech 1976; 43: 237–242. 4. Nagaraj KA. Some problems in hydrodynamic lubrication. PhD Thesis, U.S. Arizona State University, USA, 1988, pp. 108–172. 5. Nagaraj KA and Nelson HD. An analysis of gas lubricated foil journal bearings. Tribol Trans 1992; 35(1): 1–10.

6. Reddy DSK, Swarnamani S and Prabhu BS. Analyses of aerodynamic multi-leaf foil journal bearings. Wear 1997; 209: 115–122. 7. Sun D. Lecture on lubrication mechanics. Beijing: China Friendship Publishing Corporation, 1991, pp.91–94 (in Chinese). 8. Chi C. Hydromechanical lubrication. Beijing: National Defenses Industry Press, 1998, pp.485–503. 9. Heshmat H and Ku C-PR. Structural damping of selfacting compliant foil journal bearings. J Tribol 1994; 116(1): 76–82.

Appendix Notation A1i, j , A2i, j , A3i, j , A4i, j , A5i, j , A6i, j coefficients for calculating the pressure of node (i,j) C clearance of the bearing Cp contact point for the foil e eccentricity for the bearing en nominal eccentricity h film thickness of the bearing hf film thickness of the bearing generated from the oil film pressure and reaction force hfx , hfy perturbed film thickness about displacement x, y generated from the foil deformation hfx_ , hfy_ perturbed film thickness _ y_ about the velocity x, generated from the foil deformation hx , hy perturbed film thickness about displacement x, y hx_ , hy_ perturbed film thickness _ y_ about the velocity x, h dimensionless film thickness of the bearing H0 film thickness for steadystate operation L length of the bearing nx mesh number of circumferential direction nz mesh number of bearing length direction [N] reaction force matrix O center of the bearing O1, O2, . . ., O5 centers of the five separated foils. p pressure of the bearing film pa atmosphere pressure px , py perturbed pressure about displacement x, y px_ , py_ perturbed pressure about _ y_ the velocity x, p dimensionless pressure

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18 pi, j P0 Pnj R Rb Rf Se t T Tp U(i,j) x, y xO0 , yO0 z z

C T

Proc IMechE Part J: J Engineering Tribology 0(0) dimensionless pressure on node (i,j) pressure for steady operation dynamic oil film force acting on the jth foil radius of the journal radius of the bearing housing radius of the foil area of each grid time transition matrix inscribed circle’s tangent point flexibility matrix for foil deformation of node (i,j) coordinates of the axial center for the journal static equilibrium position for foil bearing axial coordinate for the bearing dimensionless axial coordinate for the bearing

0

abduction angle of the foil contact point abduction angle of the foil inscribed circle’s tangent point

’n

D H P z * * x , y * * _x , _y 1, 2 x ’

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installation angle angle between wire (xj, yj)O0 and Fy abduction angle of the foil perturbed film thickness perturbation pressure step of z direction step of direction vectors for the perturbation displacement vectors for the perturbation velocity angular coordinate for the calculation domain radius angle for the foil dimensionless bearing number viscosity of the lubrication oil circumferential angular coordinates for the bearing perturbation frequency abduction angle of the radial force attitude angle for the bearing nominal attitude angle angle*between the vector * ai and b i rotating angular speed

Performance analysis of multi-leaf oil lubricated foil bearing Liguo Hu, Guanghui Zhang, Zhansheng Liu, Ruixian Ma, Yu Wang and Jinfeng Zhang Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology published online 27 February 2013 DOI: 10.1177/1350650113475560 The online version of this article can be found at: http://pij.sagepub.com/content/early/2013/02/27/1350650113475560

Published by: http://www.sagepublications.com

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Original Article

Performance analysis of multi-leaf oil lubricated foil bearing

Proc IMechE Part J: J Engineering Tribology 0(0) 1–18 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1350650113475560 pij.sagepub.com

Liguo Hu, Guanghui Zhang, Zhansheng Liu, Ruixian Ma, Yu Wang and Jinfeng Zhang

Abstract The traditional bearing applied in the turbo-pump for the hydraulic servo system is rolling element bearing. To satisfy the demand of the high rotating speed for turbo-pump, the oil lubricated foil bearing can be employed in the rotor system. For the working liquid of the servo system is oil and the rotor for the turbo pump is submerged in the hydraulic oil, the bearing has to operate in an oil-rich environment, where the air bearing cannot be employed. The theoretical analysis and numerical simulation are carried out in this study to investigate the static and dynamic characteristics of multi-leaf oil lubricated foil bearing. For the structure form of the multi-leaf foil bearing with five symmetrical arrangements, the foil deformation equation and the Reynolds equation are solved coupled by successive over relaxation method, where the Reynolds boundary condition is employed. Then the load capacity, lift-off speed and static equilibrium position are acquired. By deriving the dynamic deformation equation of the foil, the dynamic stiffness coefficients and damping coefficients are obtained based on the perturbation method. The effect of the rotating speed and perturbation frequency on dynamic characteristics is analyzed. It indicates that the load capacity of the multi-leaf foil bearing is smaller than that of the fixed geometry oil bearing without foil deformation, whereas the stability of the bearing is increased. Keywords Multi-leaf oil lubricated foils bearing, static equilibrium position, damping coefficients, stiffness coefficients Date received: 18 September 2012; accepted: 2 January 2013

Introduction With the increased requirements for the energy density of power machinery, the rotating equipment needs to be working under harsh conditions like high rotating speed, extreme temperature and severe wear. The bearing is a key component of the high rotating speed turbo machinery. In the early 1970s and late 1960s, ASAC (Allied Signal Aerospace Co.) developed a multi-leaf cantilever gas foil bearing under the U.S. Air Force and NASA’s support to satisfy the demand of the stability and longevity for high-speed turbo machinery. Compared with the conventional bearings, the essential characteristics depend on the ﬂexible bearing surface, and the working principles belong to the elastic hydrodynamic lubrication areas. The foil structure is ﬂexible, so the gas foil bearing can establish the ﬁlm thickness according to the speed and load. The gas foil bearing can tolerate the angular deviation and a certain degree of local clearance variation generated from the misalignment of the shaft. The gas foil bearing’s structural damping

can suppress the vibration of the rotor and have excellent stability. Although the development and application of multi-leaf gas foil bearing got the powerful support from the U.S. government, the information on the bearing manufacturing process has not been published.1 In the application process, most of the lubricated medium for the multi-leaf foil bearing is gas. So the load capacity of multi-leaf gas foil bearing is insuﬃcient, which depends on the foil materials and foil manufacturing process. As the viscosity of the oil is much greater than the gas, the oil lubricated multileaf foil bearing can increase load capacity eﬀectively, School of Energy Science and Engineering, Harbin Institute of Technology, People’s Republic of China Corresponding author: Guanghui Zhang, School of Energy Science and Engineering, Harbin Institute of Technology, Main West Street, 92, Harbin 150001, People’s Republic of China. Email: [email protected]

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decrease the demands of the coating on the foils. Compared with the gas foil bearing, although the oil foil bearing has large friction loss and heat generation, it is suitable for the high speed turbo bump of hydraulic servo system, where the lubricated oil is abundant. But compared with the high speed rolling element bearing, the wear and fatigue property can be improved signiﬁcantly. Xu et al.2 introduced the design and experiment of the oil lubricated ﬁve-leaf foil bearing test-bed. The experimental results indicated that the oil lubricated foil bearing can oﬀer high rotating velocity, long life and lower friction loss. Until now, nearly no paper concerned with modeling method of this type oil lubricated foil bearing. This article focuses on the theoretical modeling of the oil lubricated foil bearing, which will make the foundation for the design of this kind bearing for turbo pump of the hydraulic servo system. The key factor of investigating the static and dynamic characteristics for multi-leaf oil foil bearing is to solve the foil deformation equation and the Reynolds equation coupled. This problem belongs to the domain of elastic hydrodynamic lubrication. As the literature about the oil lubricated foil bearing is rare, the study about the gas foil bearing is presented. Oh and Rohde3 assumed a particular shape of the gas ﬁlm, and established the foil deformation equation in the case of considering foil bending eﬀect. The friction between the foils and the friction between the foil and journal surfaces were considered. The foil deformation equation and the gas lubricated Reynolds equation was solved coupled by the ﬁnite element method. The characteristics of multi-leaf gas foil bearing including the lift-oﬀ speed and minimum gas ﬁlm thickness was obtained. Nagaraj4,5 described the geometrical structural characteristics of multi-leaf gas foil bearing in detail. The foil deformation equation and the Reynolds equation was solved and the pressure distribution, ﬁlm thickness, foil deformation was obtained. Reddy et al.6 made further study on the basis of the work of Nagaraj. The foil deformation and contact force between the adjacent foils were calculated by ﬁnite element method. Then the static equilibrium position and dynamic characteristics of gas foil bearing were presented. The ﬁve-leaf oil lubricated foil bearing is studied in this article. The structural geometry of the foil bearing is analyzed and the ﬁlm thickness expression of the clearance is obtained. Based on the work of Nagaraj,4 the foil is assumed as the cantilevered curved beam, and the reaction between the foils is point contact condition. By employing the Cartesian theorem, the deformation equation of elastic curved foil is solved. By using the Christopherson Algorithm, the ﬁlm rupture can be simulated by the Reynolds boundary condition. The Reynolds equation for oil lubricated foil bearing is solved by the ﬁnite diﬀerence scheme and successive over relaxation. By solving the foil

deformation equation and Reynolds equation coupled, the static characteristics of the bearing are got. By deriving the dynamic deformation equation of the foil, the perturbation method can be employed to solve the dynamic stiﬀness and damping coeﬃcients.

Mathematical modeling Governing equation and dimensionless form The following assumptions are employed for incompressible lubrication of journal bearing: the thickness of the oil ﬁlm is relative small compared with the length of the bearing, so the pressure gradient along the ﬁlm thickness is set as zero. The ﬂow in the bearing clearance is laminar ﬂow. The lubrication oil is isotropic, so the viscosity along the ﬁlm thickness is constant. The inertia eﬀect of the oil is ignored. The classical Reynolds equation for incompressible lubrication is as follows 1 @ h3 @p @ h3 @p 1 @h @h þ ¼ þ R2 @ 12 @ @z 12 @z 2 @ @t

ð1Þ

By introducing the dimensionless parameters h¼

h p z 6R2 12R2 , ¼ , p¼ , z¼ , 1 ¼ 2 pa C3 C pa 0:5L pa C2

The incompressible dimensionless oil lubricated Reynolds equation is as follows 2 @ 2R @ @h @h 3 @p 3 @p h h þ ¼ 1 þ 2 @ @ L @z @z @ @t ð2Þ

Differential format for Reynolds equation The oil ﬁlm pressure distribution is symmetric on middle line of the bearing, so the calculation domain is reduced to half of the bearing length, where the symmetry boundary condition is applied. The calculation domain is meshed as shown in Figure 1(a). The nodes i ¼ 1 and i ¼ nx are corresponding to the circumferential coordinates ¼ 0 and ¼ 2. The nodes j ¼ 1 and j ¼ nz are corresponding to the circumferential coordinates z ¼ 0.5L and z ¼ 0. The pressure of any arbitrary nodes (i, j), 1 4 i 4 nx, 1 4 j 4 nzþ1 is concerned with its surrounding nodes. To improve the accuracy, the nodes are inserted in the meshing element on half-step adjacent to node (i, j), which is shown in Figure 1(b). The dimensionless pressure expression on node (i, j) pi, j ¼

1 A6i, j þ 2 h_ A1i, j pi1, j þ A2i, j piþ1, j A5i, j þ A3i, j pi, j1 þ A4i, j pi, jþ1 ð3Þ

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3

(a)

(b)

Figure 1. (a) Calculation domain; (b) the half step insertion node in mesh element.

where A1i, j ¼

1 3 h 2 i1=2,j

A2i, j ¼

1 3 h 2 iþ1=2, j

A3i, j ¼

1 2R 2 3 hi, j z2 L

condition. This procedure for Reynolds boundary condition is ﬁrst proposed by Christopherson, and proved by Cryer with a strict mathematical method. As shown in Figure 1(a), the number of iterations is the number that updates the pressure in all nodes of the computational domain. So the pressure for k-th iteration is as follows pðkÞ i, j ¼

1 2R 2 ð Þ hi, j : z2 L ¼ A1i, j þ A2i, j þ A3i, j þ A4i, j

A4i, j ¼ A5i, j

A6i, j ¼

1 1 hiþ1=2, j hi1=2, j

The boundary condition and calculation procedure for Reynolds equation For the actual oil lubricated bearing, the oil ﬁlm may have rupture area where the cavitation happens. In this study, the Reynolds boundary condition is employed. In the Reynolds boundary, the oil ﬁlm starts at the maximum of the oil ﬁlm thickness, and terminates at the position where pressure derivation equaling to zero. It means that the oil ﬁlm rupture most likely occurs on the position that the pressure is low. The oil ﬁlm rupture aﬀects the pressure distribution of the bearing, which coincides with the actual situation and is called the free boundary condition. In Figure 1(a), the thick solid line O is the hint of the ﬁlm rupture position that is to meet the Reynolds boundary condition. For the exact location of O is a curve to be determined. This is an undetermined boundary condition problem. Christopherson Algorithm7 is a simple algorithm for solving the problem of the free boundary condition, and the results can satisfy Reynolds boundary

1 ðk1Þ A6i, j A1i, j pðkÞ i1, j þ A2i, j piþ1, j A5i, j ðk1Þ þ A4 p þ A3i, j pðkÞ i, j i, j1 i, jþ1

where, k ¼ 1,2 . . . is the number of iteration. Updating ðkÞ ðk1Þ pðkÞ , the expression is as follows i, j with pi, j and pi, j ðkÞ ðk1Þ pðkÞ ¼ p p þ pðk1Þ i, j i, j i, j i, j

ð4Þ

where is the relaxation coeﬃcient. For the equations set whose convergence properties is better, >1 is taken to reduce the number of iterations and accelerate convergence, which is called successive over relaxation (SOR). The SOR is employed to accelerate convergence in this article. When the kth iteration is ﬁnished, the dimensionless pressure is set to 1 for the nodes whose dimensionless pressure is smaller than 1. The following equation is considered ðkÞ pðkÞ i, j ¼ maxð1:0, pi, j Þ

ð5Þ

The convergence criterion is that the diﬀerences between two adjacent iterative calculations satisfy equation (6) Pnx Pnz ðkÞ ðk1Þ i¼1 j¼1 pi, j pi, j 5 Pnx Pnz ðk1Þ i¼1 j¼1 pi, j is usually smaller than 106.

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ð6Þ

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y Inscribed Circle

Roling Circle

y1

o1 o2

Ψ

o

x

o4

Ω

Foil

o5

Journal

In Figure 3(a), according to the geometrical relationship, the abduction angle can be obtained8 ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ h i 2 2 2 2 Rf 0:5Ro 0:25Rb Rb Ro ¼ arcsin Rf Rf Ro

Ro 2 þ arccos 1 sin 2 1 5 2 Rf ð7Þ

Bearing housing

x1

o3

The installation angle of the foil, the abduction angle T of tangent point Tp, the abduction angle C of the contact point Cp, the expressions are as follows

θ

Figure 2. The structure of five-leaf foil bearing.

The pressure distribution is obtained when the convergence criterion is satisﬁed. By employing the above point-by-point SOR iteration, the calculated pressure distribution will meet the Reynolds boundary condition.

¼ arccos

2 R2f þ R2b Rf Ro

ð8Þ

2Rf Rb

2 R2f þ Rf Ro R2b T ¼ arccos 2Rf Rf R0

ð9Þ

C ¼ T ð T Þ

ð10Þ

The foil center O1, contact point Cp and the coordinate of point p with any abduction angle of the foil are as follows

The geometry and computational domain of multi-leaf foil bearing

The geometry of multi-leaf foil bearing. The ﬁve-leaf foil bearing which is arranged symmetrically on the y-axis is analyzed in this article. Five cantilevered foils are overlapped counterclockwise as shown in Figure 2. The ﬁxed end for the foil is installed on the housing of the bearing, and the free end laps on the next foil. The inscribed circle for the foils group is tangential with each foil and a certain clearance is kept. O1, O2, . . . ,O5 are the centers of the ﬁve separated foils, which formed the rolling circle. As the geometry parameters and installation angle are same for each foil, only the foil A is studied to facilitate the analysis. Foil A is taken out from the bearing and set coordinate system x1Oy1 as the reference system, which is shown in Figure 3. The radius angle for the foil is 2p/5. In Figure 3(a), the unknown parameters in coordinate x1Oy1 are as follows: the abduction angle D of the foil, the abduction angle C of the foil contact point Cp, the abduction angle T of the foil inscribed circle’s tangent point Tp, the installation angle of the foil, the coordinate (x1O1, y1O1) of the foil center O1, the coordinate (x1p, x1p) of any point on the foil. In Figure 3(b), the unknown parameters: the coordinate (xO1, yO1) of the foil center O1, the coordinate (xO0 , yO0 ) of axial center O0 , the coordinate (xp, yp) of any point p on the foil, the ﬁlm thickness h0 on any point p on the foil.

x1O1 y1O1 x1Cp y1Cp x1p y1p

¼

¼

¼

Rf sinð Þ Rf cosð Þ Rb x1O1 y1O1

x1O1 y1O1

þ

þ

ð11Þ

Rf sinðC þ Þ Rf cosðC þ Þ

Rf sinðC þ Þ Rf cosðC þ Þ

ð12Þ

ð13Þ

In Figure 3(b), nominal radius clearance Cn ¼ R0R, the transition angle between the coordinate system xOy and x1Oy1 ¼ arctan x1Cp =y1Cp

ð14Þ

Set ¼ , then the foil center O1 in the coordinate system xOy, axial center O0 , the coordinate of any point p on the foil are as follows

xO1 yO 1

xO 0 yO 0 xp yp

x ¼ ½TðÞ 1O1 y1O1

¼

en sinðnÞ en cosðnÞ

x ¼ ½TðÞ 1p y1p

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ð15Þ

ð16Þ

ð17Þ

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(a)

(b)

Figure 3. The schematic of geometrical depiction for foil A.

where Tð Þ ¼

cos

sin

sin

cos

* * In Figure 3(b), vector a and vector b are expressed as * * a ¼ xO1 xO0 ,yO1 yO0 b ¼ xp xO0 ,yp yO0 As a foil can be treated as a part of rigid self-acting bearing, the expression of the ﬁlm thickness h0 on any point p of the foil is * h0 ¼ Rf R þ a cos ¼ Rf R * * * a b þ a * * a b

ð18Þ

Combined with the rest geometrical parameters of the foil, the foil center Oi and the coordinates of any point on the foil pi are presented in equations (19) and (20), where i ¼ 1,2,3,4,5

xOi yO i xp i yp i

x ¼ ½Tðði 1ÞÞ O1 yO 1

¼ ½Tðði 1ÞÞ

xp yp

Figure 4. The calculation domain for five-leaf foil bearing.

Division of the calculation domain The calculation domain for the ﬁve leaves foil bearing is presented in Figure 4. As the oil ﬁlm thickness is very small which can be ignored compared with the foil size, the computational domain can be expanded on the foil in the direction of . Every foil is divided with mesh size of nxnz. The area for a mesh element is Se. The foils are connected by contact points. The point on the end side of the ith foil is coincided with the point on the start side of the iþ1-th foil. The boundary condition for the whole calculation domain is as follows: For the symmetrical boundary condition

ð19Þ pð1 : nx þ 1, 1Þ ¼ p0 pð1 : nx þ 1, nz þ 1Þ ¼ pð1 : nx þ 1, nz 1Þ

ð20Þ

For the periodic boundary condition The ﬁlm thickness of any point for the foil bearing * * * ai bi h0i ¼ Rf R þ a i * * a i bi

pð0, 1 : nz þ 1Þ ¼ pðnx, 1 : nz þ 1Þ pðnx þ 1, 1 : nz þ 1Þ ¼ pð1, 1 : nz þ 1Þ

ð21Þ Also the Reynolds boundary condition will be satisﬁed by the Christopherson Algorithm.

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Figure 6. The schematic of foil forces analysis.

The generalized virtual force F 0 is set as zero, so the radial displacement j for node j under the radial force F can be expressed as follows Figure 5. The schematic of foil deformation analysis.

Z

Z 0d þ

j ¼

0

The solving of the foil deformed flexibility matrix and reaction force The pressure in the clearance of the multi-leaf foil bearing can generate the bearing load capacity, and it will make the foil deform. The clearance will be changed and aﬀect the pressure distribution. The deformation of the foil with oil ﬁlm pressure and interaction force will make the foundation for the analysis of the bearing. In this article, the foil is assumed as the cantilevered curved beam and the ﬂexibility matrix for the curved beam will be obtained. By analyzing the forces for the foil bearing, the reaction forces between foils can be solved. The flexibility matrix of the foil deformation. The foil can be simpliﬁed as the cantilevered curved beam in Figure 5. The width of the beam is shown in Figure 4, where the area is formed by the dotted line. The radial force F acting on the node i with abduction angle of will result in the radial displacement j for the node j with arbitrary abduction angle of . With the assumption of small deformation, the Castigliano’s theorem can be employed in this article. For the case of 4, the radial force and the generalized virtual force F 0 for j node with abduction angle of act on the foil simultaneously Mð Þ ¼ F Rf sinð Þ Mð Þ ¼ F Rf sinð Þ þF 0 Rf sinð ð ÞÞ

2 ½0, Þ For the curve i j 2 ½ , For the curve j A

Radial deformation j is Z

j ¼ 0

@Mð Þ Mð Þ Rf d @F0 EI

¼

F R3f 4EI

F Rf sinð Þ Rf sinð ð ÞÞ Rf d EI

½2 cosð Þ þ sinð Þ

sinð þ Þ 2 ½0,

Similarly, for the case of 44, the radial displacement is expressed

j ¼

F R3f

½2 cosð Þ þ sinð Þ 4EI sinð þ Þ 2 ½,

The ﬂexibility matrix U(i, j) indicates the displacement of node j with unit force acting on the node i for the cantilevered beam. Then the ﬂexibility matrix is as follows

Uði, jÞ ¼

8 > R3f > > ½2 cosð Þ þ sinð Þ > > > 4EI > > > > < sinð þ Þ 14j4i > > R3f > > > ½2 cosð Þ þ sinð Þ > > 4EI > > > : sinð þ Þ i4j4n

Reaction force between the foils. The foils are contacted with each other at contact point, as shown in Figure 6. The radial displacements of two foils at the contact point are the same. By ignoring the friction force, the interaction force is only radial reaction force. The dynamic oil ﬁlm force acting on the jth foil is Pnj ¼ pjSe. The reaction force on the jth foil generated from the j1 foil is Ni. The reaction force on the jth foil generated from the jþ1 foil is Niþ1. The forces mentioned above are balanced on the j the foil.

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Combing the ﬂexibility matrix, the following equation can be derived Niþ1 Uðn, 1Þ þ Pnj ðkÞUðk, 1Þ þ Ni Uð1, 1Þ ¼ Ni Uðn, nÞ þ Pnj1 ðkÞUðk, nÞ þ Ni1 Uð1, nÞ where k ¼ 1, 2, . . . , n. Further Ni ½Uð1, 1Þ þ Uðn, nÞ Niþ1 Uðn, 1Þ Ni1 Uð1, nÞ ¼ Pnj1 ðkÞUðk, nÞ Pnj ðkÞUðk, 1Þ ð22Þ For ﬁve-leaf foil bearing, it can be written in matrix form ½A½N ¼ ½Pn½Uc

Figure 7. The schematic of the static equilibrium position for the shaft in foil bearing.

ð23Þ

where ½N ¼ ðN1 , N2 , N3 , N4 , N5 ÞT

ð24Þ

0

Uð1, 1Þ þ Uðn, nÞ Uðn, 1Þ 0 B Uð1, nÞ Uð1, 1Þ þ Uðn, nÞ Uðn, 1Þ B B B ½A ¼ B 0 Uð1, nÞ Uð1, 1Þ þ Uðn, nÞ B 0 0 Uð1, nÞ @ Uðn, 1Þ 0 0 pnz 1ð1Þ pnz 1ðnÞ pnz 5ð1Þ B B pnz 2ð1Þ pnz 2ðnÞ pnz 1ð1Þ B ½Pn ¼ Se B B pnz 3ð1Þ pnz 3ðnÞ pnz 2ð1Þ B @ pnz 4ð1Þ pnz 4ðnÞ pnz 3ð1Þ pnz 5ð1Þ pnz 5ðnÞ pnz 4ð1Þ T ½Uc ¼ U1, 1 , . . . , Un, 1 , U1, n , . . . , Un, n

0

The support reaction force array [N] can be obtained by solving equation (23), where the interaction forces between foils are calculated. Expression of the film thickness. From equation (18), the ﬁlm thickness of the foil bearing with foil deformation is h ¼ h0 þ hf

ð25Þ

where hf is the radial deformation from the ﬁlm pressure and the reaction force on the foil, namely, the deformation of the point i on the jth foil hf jðiÞ ¼ Nj Uð1, iÞ Njþ1 Uðn, iÞ n X Pnj ðkÞ Uðk, iÞ þ k¼1

pnz 5ðnÞ

0 0

Uð1, nÞ 0

Uðn, 1Þ Uð1, 1Þ þ Uðn, nÞ

0 Uðn, 1Þ

Uð1, nÞ

Uð1, 1Þ þ Uðn, nÞ

1

1 C C C C C C A

C pnz 1ðnÞ C C pnz 2ðnÞ C C C pnz 3ðnÞ A pnz 4ðnÞ 5ð2nÞ

where i ¼ 1, 2 . . . ,n, j ¼ 1, 2 . . . 5. Njþ1 ¼ N1 when j ¼ 5.

The load capacity and static equilibrium position for the foil bearing As the structure of the ﬁve-leaf foil bearing is not symmetrical, the pressure distribution for the bearing depends on the eccentricity e and the attitude angle ’. The variables mentioned above are demanded to obtain the static equilibrium position simultaneously. As shown in Figure 7, the following relationship needs to be satisﬁed Fy ¼ W

ð26Þ

Fx ¼ 0

ð27Þ

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The angle 0 between the wire (xj,yj)-O0 and Fy is deﬁned as follows

H 0 ¼ h0 þ hf

ð34Þ

0 ¼ arctan yi yO0 , xi xO0

Hx ¼ hx þ ivhx_

ð35Þ

Then

Hy ¼ hy þ ivhy_

ð36Þ

Z2 Z0 p cos R dzd

Fy ¼ 2

ð28Þ

0 L=2

Z2 Z0 Fx ¼ 2

p sin R dzd

ð29Þ

H0 is the ﬁlm thickness for steady-state operation. H is the perturbed ﬁlm thickness. The perturbed ﬁlm pressure P can be expanded by the Taylor equation about the steady-state operated pressure P0 * * * * p ¼ P0 þ px x þ py y þ px_ x_ þ py_ y_

0 L=2

The pressures p (i, j) on each node have been obtained in the section ‘The boundary condition and calculation procedure for Reynolds equation’.

Dynamic characteristics of the multi-leaf foil oil lubricated bearings In the study of the dynamic coeﬃcients for the multileaf foil bearing, the elastic deformation of the foil and the ﬁlm pressure are coupled. Thus, the perturbation of the pressure distribution will cause the perturbation deformation of the foil, and then aﬀect the perturbation of the ﬁlm pressure. So the perturbation method will be employed in this study to derive the expression of the perturbed foil deformation. By coupling with the perturbed Reynolds equation, the stiﬀness and damping coeﬃcients can be obtained. The perturbation of the foil deformation. If the journal perturbs around the static equilibrium position (xO0 , yO1) of the axial center O0 when the perturbation frequency is u rad/s, the position of the axial center can be expressed as

¼ P0 þ P ¼ P0 þ Px xeivt þ Py yeivt

ð37Þ

where Px ¼ px þ ivpx_

ð38Þ

Py ¼ py þ ivpy_

ð39Þ

Taking equation (21) into equation (25), the following equation can be obtained * * H ¼ h0 þ hf ¼ x sin y cos * * þ Hf x x_ þ Hf y y_ ¼ sin þ Hf x xeivt þ cos þ Hf y yeivt ð40Þ where Hf x ¼ hf x þ ivhf x_

ð41Þ

Hf y ¼ hf y þ ivhf y_

ð42Þ

* x ¼ xO0 þ x ¼ xO0 þ xeivt

ð30Þ

Comparing equation (40) with equation (33)

* þ y ¼ yO0 þ yeivt

ð31Þ

Hx ¼ sin þ Hf x

ð43Þ

Hy ¼ cos þ Hf y

ð44Þ

y ¼ yO 0

* * where x and y are the vectors of the perturbation displacement; x and y are the amplitudes of the vectors * * x_ ¼ ivxeivt ¼ iv x * * y_ ¼ ivyeivt ¼ iv y

ð32Þ

Combined with equation (25), the Taylor expansion of the perturbed ﬁlm thick function h for steady-state operation can be obtained * * * * h ¼ h0 þ hf0 þ hx x þ hy y þ hx_ x_ þ hy_ y_ ¼ H0 þ H ¼ H0 þ Hx xeivt þ Hy yeivt

* * where is the angle between vector ai and bi as shown in Figure 3. As the dotted line range of the calculation domain in Figure 4 has been assumed as the bending beam, the pressure and foil deformation on the nodes can be expressed with the matrix form as follows

T ½pz ¼ p1, nz , p2, nz , . . . , pi, nz , . . . , pnxþ1, nz

ð45Þ

T hf ¼ hf 1, hf 2, . . . , hf i, . . . , hf nx þ 1

ð46Þ

ð33Þ

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By employing equation (23)

along the z-axis is ignored, equation (54) can be simpliﬁed as follows

½N ¼ ½A1 ½Pn½Uc ¼ ½B5ðnxþ1Þ ½pz

ð47Þ

The matrix [B] can be obtained by merging Pnz in equation (23). Substitute equation (47) into equation (25), then hf can be expressed by matrix [Pz] hf ¼ ½UHEðnxþ1Þðnxþ1Þ ½pz

ð48Þ

If the dynamic deformation coeﬃcients matrix [UHE] can be obtained, the [hf] can be got as the matrix [pZ] is known. hf ¼ hf0 þ hf * * ¼ hf0 þ Hfx x þ Hfy y * ¼ ½UHE½Pz0 þ ½UHE½Pzx x * þ ½UHE Pzy y

ð49Þ

Hfx ¼ ½UHE½Pzx

ð50Þ

Hfy ¼ ½UHE Pzy

ð51Þ

The matrix [Pzx], [Pzy] are the partial derivative of * * pressure tox , y . Substituting equations (50) and (51) into equation (40), then the column array of the perturbation deformation for each node can be obtained

½H ¼ sin½ þ Hf x xeivt

þ cos½ þ Hf y yeivt

ð52Þ

The discretization and solution of the perturbed Reynolds equation. Substituting equations (33) and (37) into the Reynolds equation (equation (1)), the following equation can be got by ignoring the second or higher order minute amount.

@ @P0 @ @P0 @h0 H30 H30 þ R2f ¼ 6R2f @ @z @ @z @

@H @H þ 12R2f @ @t

ð55Þ

@H0 @ @2 @2 þ H30 2 þ R2f H30 2 @ @ @ @z

@P0 @H0 @2 P0 þ 3H20 2 @ @ @ 2 @ P0 þ 3R2f H20 2 iv 12R2f @z

ð56Þ

G2 ¼ 6H0

@P0 6R2f @

ð57Þ ð58Þ

Substitute equations (37) and (40) of P, H into the above equations, the following can be obtained G1 Px þ G2 Hf x þ G3

G1 Py þ G2 Hf y þ G3

@Hf x ¼ G2 sin G3 cos @ ð59Þ @Hf y ¼ G2 cos G3 sin @ ð60Þ

Substitute equation (50) into equation (59), then the perturbed Reynolds equation for any node ði, jÞ on the calculation domain is as follows G1 Px ði,jÞ þ TEMðiÞ ½Pz x ¼ G2 sin ðiÞ G3 cos ðiÞ ð61Þ where TEMðiÞ ¼ G2 UHEði, 1 : nx þ 1Þ 1 G3 ½UHEði þ 1, 1 : nx þ 1Þ 2

ð53Þ UHEði 1, 1 : nx þ 1Þ

@ 3 @P 2 @P0 H0 þ 3H0 H @ @ @

2 @ 3 @P 2 @P0 þ Rf H0 þ 3H0 H @z @z @z ¼ 6R2f

G1 ¼ 3H20

þ

@H ¼0 @

where

G3 ¼ 3H20

Then

G1 P þ G2 H þ G3

ð62Þ

The ﬁrst right hand term of equation (62) can be discrete by the ﬁnite diﬀerence method as follows G1 Px ði, jÞ ¼ A1 ði, jÞ Px ði 1, jÞ þ A2 ði, jÞ Px ði, jÞ ð54Þ

Equation (53) is the steady-state operation Reynolds equation. Equation (54) is the perturbed Reynolds equation. As the ﬁlm thickness variation

þ A3 ði, jÞ Px ði þ 1, jÞ þ A4 ði, jÞ Px ði, j 1Þ þ A5 ði, jÞ Px ði, j þ 1Þ ¼ A2 ði, jÞ Px ði, jÞ þ TXði, jÞ

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ð63Þ

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where

For the oil ﬁlm rupture area

1 H3 ði, jÞ 2 0 3 H2 ði, jÞ ½H0 ði þ 1, jÞ H0 ði 1, jÞ 4 2 0 A2 ði, jÞ ¼ A1 ði, jÞ A3 ði, jÞ A4 ði, jÞ A5 ði, jÞ 1 3 H3 ði, jÞ þ H2 ði, jÞ ½H0 ði þ 1, jÞ A3 ði, jÞ ¼ 2 0 4 2 0 H0 ði 1, jÞ 2 Rf A4 ði, jÞ ¼ A5 ði, jÞ ¼ H30 ði, jÞ z

A1 ði, jÞ ¼

For the case of j 6¼ nz, substituting equation (63) into equation (61)

Px ði, jÞ ¼ Py ði, jÞ ¼ 0 To solve Px(i,j) and Py(i,j), SOR method is used in this article. Dynamic stiffness and damping coefficients. As shown in equations (38) and (39), the complex variable Px(i,j) and Py(i,j) can be obtained, where the real and imaginary part are respectively the derivative of the ﬁlm pressure to the journal displacement and velocity Px ði, jÞ ¼ px ði, jÞ þ iv px_ ði, jÞ Py ði, jÞ ¼ py ði, jÞ þ iv py_ ði, jÞ Then

TXði, jÞ þ TEMðiÞ ½Pzx þG2 ði, jÞ sin ðiÞ þ G3 ði, jÞ cos ðiÞ Px ði, jÞ ¼ A2 ði, jÞ

ð64Þ

For the case of j ¼ nz, the coeﬃcient of Px(i,nz) is deﬁned as E(xi) EðiÞ ¼ A2 ði, nzÞ þ G2 ði, nzÞ UHEði, iÞ 1 G3 ði, nzÞ ½UHEði þ 1, iÞ UHEði 1, iÞ þ 2 ¼ A2ði, nzÞ þ EFðiÞ TXði, jÞ þ TEMðiÞ ½Pzx þ G2 ði, jÞ sin ðiÞ Px ði, jÞ ¼ EðiÞ þG3 ði, jÞ cos ðiÞ EFðiÞ Pzx ði nzÞ EðiÞ

The Py(i,j) on the point (i,j) can also be expressed as the equation mentioned above. The solving procedure for Px(i,j) and Py(i,j) needs to satisfy the cyclic boundary condition as follows, which is shown in Figure 4 Px ð0, 1 : nzÞ ¼ Px ðnx, 1 : nzÞ

KDxx

Px

¼ 2 0

Py

0

sin Rf ddz cos

L=2

Z2 Z0 ¼ 2

KDyx

Z2 Z0

sin cos

Rf ddz

L=2

Where the matrix [KD] is as follows ½KD ¼

Kxx þ iv Dxx

Kxy þ iv Dxy

Kyx þ iv Dyx

Kyy þ iv Dyy

C0 Re½KD K ¼ W C0 Im½KD D ¼ Wv The matrix K and D are dimensionless form of the stiﬀness and damping matrix.

Numerical simulations Calculation of the foil deformation

Py ð0, 1 : nzÞ ¼ Py ðnx, 1 : nzÞ Px ðnx þ 1, 1 : nzÞ ¼ Px ð1, 1 : nzÞ Py ðnx þ 1, 1 : nzÞ ¼ Py ð1, 1 : nzÞ Symmetric boundary conditions Px ð1 : nx, nz þ 1Þ ¼ Px ð1 : nx, nz 1Þ Py ð1 : nx, nz þ 1Þ ¼ Py ð1 : nx, nz 1Þ Fixed value boundary conditions Px ð0 : nx þ 1, 1Þ ¼ 0

KDxx KDyx

Px ð0 : nx þ 1, 1Þ ¼ 0

To verify the accuracy of the foil deformation calculation with the Castigliano’s theorem, the comparison between the Castigliano’s theorem and the ﬁnite element method is presented. The ﬁnite element method is carried out by the software Abaqus. The simulation parameters are as follows: ¼ 80 , ¼ 60 , Rf ¼ 10 mm, the width of the curved beam is 1 mm, thickness of the beam section is 0.5 mm. The curved beam is divided into eight nodes. The results are presented in Table 1. It indicates that the results are fundamental consistence. It means that the Castigliano’s theorem can be employed to solve the ﬂexibility matrix U(i, j) accurately.

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Table 1. The result of the radial displacement for curved beam. Radial deformation (mm) Node numbering

Abduction angle ( )

Castigliano theorem results

Finite element method results

1 2 3 4 5 6 7 8

10 20 30 40 50 60 70 80

0.00579737 0.0220320 0.0465028 0.0764881 0.108888 0.140385 0.168021 0.190551

0.00582508 0.0220927 0.0465999 0.0766226 0.109059 0.140588 0.168201 0.190704

Table 2. The parameters of five multi-leaf foil bearing. Item

Description

Value

NF Rf TI E R Rb Ro L W P0 nx nz

Number of the foil Radius of the foil (m) Thickness of the foil (m) Elastic modulus of the foil (Pa) Radius of the journal (m) Radius of the bearing housing (m) Radius of the tangent circle (m) Length of the bearing (m) Bearing load capacity (N) Viscosity of the lubricated oil (Pa s) Ambient pressure (MPa) The number of the element for the foil along circumferential direction The number of the element for the foil along axial direction

5 0.01 0.0005 2.11011 0.007 0.008 0.00705 0.01 1.5 0.59610-3 0.010325 51 21

The static characteristics of oil-lubricated foil bearing The foil structure is shown in Figure 1, and the detail parameters are presented in Table 2. The static characteristics of the ﬁve-leaf oil foil bearing are simulated. The comparisons between the cases with and without foil deformation are presented. With the parameters in Table 2, the nominal eccentricity is 0.02103 mm, and the nominal attitude angle is 20 . The rotating speed is 10,000 r/min. As shown in Figure 2, the ﬁve leaves are completely identical. In Figure 8(a), the three dimensionless pressure distribution of the foil bearing are presented where foil deformation is considered. In Figure 8(b), when the foil deformation is ignored, the center of the journal O0 is close to the foil A and E. The minimum ﬁlm thickness locates between the foil A and the journal, which is equaling to 0.62 C0 in Figure 9. The maximum pressure is on the foil A, and it is 5.9 P0. When the foil deformation is considered, the minimum ﬁlm thickness is still near the foil A, but increased to 0.76

C0 that is presented in Figure 10. Similar to the case of ignoring the foil deformation, the maximum pressure is still on the foil A, but decreased to 4.8 P0. The pressure of the bearing with the foil deformation considered is higher than that of ignoring the foil deformation in general. As the maximum pressure happens on the foil A and foil E, the maximum deformation of the foil is on the foil A, which is shown in Figure 10. The maximum foil deformation of the bearing is 0.14 C0, which is shown in Figure 11. For the cases with and without the foil deformation, the maximum ﬁlm thickness happens on the overlap joint of adjacent foils. From the numerical simulation, it indicates that for the area of maximum ﬁlm thickness, the pressure of the ﬁlm is a little higher than that of the atmosphere pressure. So the rupture area of the oil ﬁlm is relative small. When the foil deformation is considered, the domain of higher oil ﬁlm pressure increases on foil A and foil E. The oil ﬁlm rupture area on foil E reduces obviously. It means that the foil deformation can enlarge the pressure area of the bearing, and reduce the area of

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Figure 8. The pressure distribution for oil lubricated foil bearing: (a) pressure distribution for the foil bearing with foil deformation; (b) the comparison of the pressure distribution for foil’s middle line with the case of considering and ignoring the deformation.

Figure 9. The film thickness distribution ignoring the deformation of the foil.

Figure 10. The film thickness distribution considering the deformation of the foil.

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ﬁlm rupture. So the stability of the bearing is improved when the foil deformation happens. As the maximum pressure decreases, so the load capacity of the bearing with foil deformation is smaller than that of without foil deformation. For the structure model in Table 2, the lift oﬀ speed for the bearing is studied in this article. By numerical simulation, when the rotating speed is smaller than 970 r/min, the bearing load capacity is smaller than 1.5 N. So the rotating speed of 970 r/min is the case that the shaft and the foil has been separated by the lubricated oil, which is called the lift oﬀ speed. As the rotating speed increases to 100,000 r/min, the static equilibrium position locus of the shaft center is depicted in Figure 12. The variation of the nominal attitude angle with the nominal eccentricity is presented in Figure 13. In Figure 12, as the rotating speed increases, the center of the shaft is gradually approaching to the center of the bearing, where the nominal eccentric ratio decreases. But varied range of the nominal attitude angle is about 6 . The locus of the shaft center moving to the bearing center is nearly the straight line. This phenomenon is generated from the symmetrical arrangement and elastic deformation of the foils, which leads to the motion of the shaft along the load direction. The displacement of the shaft perpendicular to the load direction is relative small. So the force of direction ’n is small, where the destabilizing force of the oil ﬁlm is small. It indicates that the ﬁve-leaf foil bearing has an excellent stability. In Figure 14 the variation of the nominal Sommerfeld coeﬃcient with the nominal eccentricity is presented, where the Sommerfeld coeﬃcient is deﬁned as follows Sn ¼ RLðR=C0 Þ2 =W

O

90

0.0 0.2

75

0.4 60

0.6 0.8

45

1.0

30 0

15

Nominal attitude angle ϕ n

Figure 12. The orbit of static equilibrium position for five multi-leaf foil bearing.

Figure 13. The relation between nominal attitude angle and nominal eccentricity with the balancing of shaft center.

Figure 11. The film thickness distribution considering the deformation of the foil.

Figure 14. The relation between the nominal Sommerfeld coefficients and nominal eccentricity.

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Figure 15. The stiffness coefficients varying with the nominal Sommerfeld coefficient.

As the eccentricity increases, the Sommerfeld coefﬁcient decreases. When the eccentricity is small, the variation of the Sommerfeld is severe. It indicates that the shaft center position is stable when the rotating speed is bigger. When the rotating speed is bigger than 10,000 r/min, the eccentricity ratio is 0.2, and the Sommerfeld coeﬃcient is larger than 20. It indicates that the designed ﬁve-leaf foil bearing is suitable for the high rotating speed operation. When the perturbation frequency ratio p ¼ / is equaling to 1 and the speed ranges from 971 to 10,000 r/min, the variation of the stiﬀness coeﬃcients with the nominal Sommerfeld coeﬃcient is analyzed in Figure 15. For the heavy load case, the main stiﬀness on load direction Kyy is slightly bigger than that of Kxx . For the case of high rotating speed and light load capacity, the main stiﬀness of the two direction increases simultaneously as the rotating speed increases. The diﬀerence between the Kyy and Kxx decreases gradually. But Kyy is consistently bigger than Kxx . The cross coupling stiﬀness Kxy is negative when the Sommerfeld coeﬃcient is smaller than 0.5. It varied dramatically when the Sommerfeld coeﬃcient is 0.5. As the Sommerfeld coeﬃcient increases, the Kxy changes to positive value. The cross coupling stiﬀness Kyx is always negative. In Figure 16, the dimensionless damping coeﬃcients varying with the nominal Sommerfeld coeﬃcient are presented, where the perturbation frequency is 1. When the Sommerfeld coeﬃcient is small, the diﬀerence between the two main damping coeﬃcients is obvious. The main damping coeﬃcient Dyy is bigger than the coeﬃcient Dxx . But as the Sommerfeld coeﬃcient increases, the two main damping coeﬃcients increase and the diﬀerence between

them disappears. When the rotating speed is lower than 15,000 r/min, the cross coupling damping coeﬃcients are all negative. They ﬂuctuate as the Sommerfeld coeﬃcient increases, and the tendency is similar. When the Sommerfeld coeﬃcient is in the domain between 2.5 and 3.6, the damping coeﬃcients Dxy and Dyx are changed into the positive value. The coeﬃcient Dyx increases as the Sommerfeld coeﬃcient increases, and the coeﬃcient Dxy becomes negative the absolute value approaches the coeﬃcient Dxy . The variation of the dimensionless stiﬀness and damping coeﬃcients with the perturbation frequency for rotating speed of 3000 r/min, 15,000 r/min and 50,000 r/min are presented in Figures 17 to 20. As shown in Figure 17, the main stiﬀness coeﬃcient Kyy is bigger than Kxx all along. But the diﬀerence between them is neglect able when the rotating speed approaches 50,000 r/min. As the rotating speed is lower at about 3000 r/min, the main stiﬀness coeﬃcients are not insensitive to the variation of the perturbation frequency. When the rotating speed increases to 15,000 r/min, the main stiﬀness coeﬃcients increase as the perturbation frequency increases. This phenomenon can be observed obviously for the case of 50,000 r/min. It can be concluded that the perturbation frequency aﬀects the main stiﬀness coeﬃcients. As the perturbation increases, the main stiﬀness coeﬃcients increase. It indicates that the perturbation frequency ratio has little eﬀect on the cross coupling stiﬀness, which is shown in Figure 18. Even the rotating speed reaches about 50,000 r/min, the variation range for dimensionless cross coupling stiﬀness coeﬃcient is still small. As the perturbation frequency ratio increases, the cross coupling stiﬀness coeﬃcient Kxy decreases,

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Dimensionless damping coefficients

Hu et al.

1

0.1

0.01

D xx D yy D xy positive D xy negative D yx positive

1E-3

D yx negative 10 Nominal Sommerfeld coefficient S n =

μ Ω RL ⎛ R ⎞ ⎜ ⎟ π W ⎝ C0 ⎠

100

2

Dimensionless main stiffness coefficients

Figure 16. The damping coefficients varying with the nominal Sommerfeld coefficient.

40

K yy 3 krpm K xx 3 krpm

35

K yy 15 krpm K xx 15 krpm

30

K yy 50 krpm K xx 50 krpm

50 krpm

25 20 15

15 krpm

10 3 krpm

5 0 0

2

4 6 Perturbation frequency ratio ν / Ω

8

10

Figure 17. The main stiffness coefficients varying with perturbation frequency.

but the coeﬃcient Kyx increases. The dimensionless damping coeﬃcients varying with the perturbation frequency are presented in Figures 19 and 20. The inﬂuence of the frequency perturbation ratio on the dimensionless damping coeﬃcients is apparent when the rotating speed is high. When the frequency perturbation ratio increases, the main damping coeﬃcients Dxx and Dyy decrease. The cross coupling stiﬀness coeﬃcient Dyx decreases and the Dxy increases. The above tendency is not obvious when the rotating speed is low. The above simulation indicates that the perturbation frequency has little inﬂuence on the stiﬀness

and damping coeﬃcients when the rotating speed is low. As the rotating speed is high, the main stiﬀness and damping coeﬃcients changes obviously with the perturbation frequency. As the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcients decreases. The cross-coupling stiﬀness and damping coeﬃcients change little with the perturbation frequency. This phenomenon is similar to the experimental results of gas bump foil bearing in Heshmat and Ku.9 In their study, as the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcient decreases.

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2.5

K xy 3 krpm K yx 3 krpm K xy 15 krpm K yx 15 krpm

50 krpm

2.0 1.5

15 krpm

Dimensionless coupling stiffness coefficients

1.0

K xy 50 krpm K yx 50 krpm

0.5 3 krpm

0.0

3 krpm

–0.5 –1.0

15 krpm

–1.5 –2.0

50 krpm

–2.5 0

2

4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 18. The cross coupling stiffness coefficients varying with perturbation frequency.

50 krpm

5.5

D yy 3 krpm D xx 3 krpm

5.0

D yy 15 krpm D xx 15 krpm

4.5 Dimensionless main damping coefficients

4.0

D yy 50 krpm

3.5

D xx 50 krpm

3.0 15 krpm

2.5 2.0 1.5 1.0 0.5

3 krpm

0

2 4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 19. The main damping coefficients varying with perturbation frequency.

0.15 50 krpm

D xy 3krpm D yx 3krpm

0.10

D xy 15 krpm D yx 15 krpm

15 krpm

0.05

Dimensionless coupling damping coefficients

D xy 50 krpm D yx 50 krpm

15 krpm

0.00

3 krpm

–0.05 –0.10 50 krpm

–0.15 0

2

4 6 8 Perturbation frequency ratio ν / Ω

10

Figure 20. The cross coupling damping coefficients varying with perturbation frequency.

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Conclusions The geometrical structure of the multi-leaf foil bearing with ﬁve symmetrical arranged foils are analyzed in this article. By considering the deformation of the foil, the steady-state incompressible Reynolds equation is solved. The static characteristics such as the lift-oﬀ speed and static equilibrium position are obtained, which make the foundation for the dynamic characteristics of the oil foil bearing. The relation between the perturbed pressure and foil deformation is derived by the perturbation method and the dynamic deformation equation is obtained. By employing the SOR method and combining the boundary condition, the perturbed Reynolds equation is solved. By integrating the pressure, the dynamic parameters including the stiﬀness and damping coefﬁcients concerned with the perturbation is got. The variation of the stiﬀness and damping coeﬃcients with the Sommerfeld parameter is analyzed. The inﬂuence of the perturbation frequency on the dynamic coeﬃcients is presented. As the perturbation frequency increases, the main stiﬀness coeﬃcient increases and the main damping coeﬃcient decreases. This phenomenon is obvious for the case of the high rotating speed. For the cross coupling stiﬀness and damping coeﬃcients, the perturbation frequency has little inﬂuence. The study makes the foundation for the design and analysis of the oil foil bearing for turbo pump, which is employed in the hydraulic servo system. The experiment needs to be carried out in further study to verify the accuracy of the modeling method for oil lubricated foil bearing.

Funding This study was supported by the National Natural Science Foundation of China (grant no. 51106035) and China Postdoctoral Science Foundation funded project (grant no. 2012M510088).

References 1. DellaCorte C, Radil KC, Bruckner RJ, et al. Design, fabrication, and performance of open source generation I and II compliant hydrodynamic gas foil bearings. Tribol Trans 2008; 51(3): 254–264. 2. Xu HJ, Liu ZS, Zhang GH, et al. Design and experiment of oil lubricated five-leaf foil bearing test-bed. J Eng Gas Turb Power 2009; 131: 054505. 3. Oh KP and Rohde SM. A theoretical investigation of the multileaf journal bearing. J Appl Mech 1976; 43: 237–242. 4. Nagaraj KA. Some problems in hydrodynamic lubrication. PhD Thesis, U.S. Arizona State University, USA, 1988, pp. 108–172. 5. Nagaraj KA and Nelson HD. An analysis of gas lubricated foil journal bearings. Tribol Trans 1992; 35(1): 1–10.

6. Reddy DSK, Swarnamani S and Prabhu BS. Analyses of aerodynamic multi-leaf foil journal bearings. Wear 1997; 209: 115–122. 7. Sun D. Lecture on lubrication mechanics. Beijing: China Friendship Publishing Corporation, 1991, pp.91–94 (in Chinese). 8. Chi C. Hydromechanical lubrication. Beijing: National Defenses Industry Press, 1998, pp.485–503. 9. Heshmat H and Ku C-PR. Structural damping of selfacting compliant foil journal bearings. J Tribol 1994; 116(1): 76–82.

Appendix Notation A1i, j , A2i, j , A3i, j , A4i, j , A5i, j , A6i, j coefficients for calculating the pressure of node (i,j) C clearance of the bearing Cp contact point for the foil e eccentricity for the bearing en nominal eccentricity h film thickness of the bearing hf film thickness of the bearing generated from the oil film pressure and reaction force hfx , hfy perturbed film thickness about displacement x, y generated from the foil deformation hfx_ , hfy_ perturbed film thickness _ y_ about the velocity x, generated from the foil deformation hx , hy perturbed film thickness about displacement x, y hx_ , hy_ perturbed film thickness _ y_ about the velocity x, h dimensionless film thickness of the bearing H0 film thickness for steadystate operation L length of the bearing nx mesh number of circumferential direction nz mesh number of bearing length direction [N] reaction force matrix O center of the bearing O1, O2, . . ., O5 centers of the five separated foils. p pressure of the bearing film pa atmosphere pressure px , py perturbed pressure about displacement x, y px_ , py_ perturbed pressure about _ y_ the velocity x, p dimensionless pressure

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18 pi, j P0 Pnj R Rb Rf Se t T Tp U(i,j) x, y xO0 , yO0 z z

C T

Proc IMechE Part J: J Engineering Tribology 0(0) dimensionless pressure on node (i,j) pressure for steady operation dynamic oil film force acting on the jth foil radius of the journal radius of the bearing housing radius of the foil area of each grid time transition matrix inscribed circle’s tangent point flexibility matrix for foil deformation of node (i,j) coordinates of the axial center for the journal static equilibrium position for foil bearing axial coordinate for the bearing dimensionless axial coordinate for the bearing

0

abduction angle of the foil contact point abduction angle of the foil inscribed circle’s tangent point

’n

D H P z * * x , y * * _x , _y 1, 2 x ’

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installation angle angle between wire (xj, yj)O0 and Fy abduction angle of the foil perturbed film thickness perturbation pressure step of z direction step of direction vectors for the perturbation displacement vectors for the perturbation velocity angular coordinate for the calculation domain radius angle for the foil dimensionless bearing number viscosity of the lubrication oil circumferential angular coordinates for the bearing perturbation frequency abduction angle of the radial force attitude angle for the bearing nominal attitude angle angle*between the vector * ai and b i rotating angular speed