Stability analysis of hydrodynamic bearing with herringbone grooved sleeve

Tribology International 55 (2012) 15–28

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Stability analysis of hydrodynamic bearing with herringbone grooved sleeve Shih-Kang Chen, Hsien-Chin Chou, Yuan Kang n Department of Mechanical Engineering, Chung Yuan Christian University, Chung-Li 320, Taiwan, ROC

a r t i c l e i n f o

abstract

Article history: Received 12 July 2011 Received in revised form 18 May 2012 Accepted 21 May 2012 Available online 5 June 2012

This study presents a numerical method to investigate the stability analysis of a hydrodynamic journal micro-bearing. The governing dimensionless Reynolds equation is solved by using the finite difference method. The Green’s theorem is utilized to deal with the discontinuity of the oil-film thickness, and the Gaussian elimination method is used to solve the simultaneous discrete equations. Optimal values for various design parameters are obtained to maximize the load capacity and to improve the stability characteristics. The accomplishments of this study will help the designers who deal with the grooved micro-bearing to select the design parameters to satisfy the required characteristics. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Hydrodynamic journal micro-bearing Reynolds equation Finite difference method Stability analysis

1. Introduction The herringbone grooved journal bearing (HGJB) is designed for micro-spindle in order to improve the lubrication characteristics and it gradually replaces the early design of plain journal bearing (PJB) used in the storage media, such as CD, VCD, and DVD. However the bearing characteristics and spindle stability are always worse than those of a PJB except the HGJB operates under low eccentricity conditions. This deterioration is caused by the saw-toothed pressure distribution generated by the grooves located on the internal surfaces of a HGJB. The narrow groove theory (NGT) was used to analyze the HGJB by early researchers; it is assumed that the number of grooves in the bearing is infinite and the pressure distribution induced by the grooves is simplified as the mean value of the saw-toothed pressure distribution, which is higher than that in a PJB. Hirs [1] applied the NGT to obtain the pressure distribution, friction moment, and stability of a HGJB. Bootsma [2,3] investigated load capacity and stability of a HGJB for different shapes of spiral grooves including spherical and conical profiles by the NGT. They both compared the analysis results with experimental data. Kang et al. [4] applied the finite difference method with the ¨ help of conservation of mass, Grumbel boundary conditions, and coordinate transformation to analyze the static and dynamic characteristics of a HGJB. The results showed that the stability of a PJB at low eccentricity can be improved by applying either rectangular or circular grooves. Jang and Yoon [5,6] analyzed the difference of static and dynamic characteristics between a grooved bearing with plain journal (GBPJ) and a plain bearing with grooved

n

Corresponding author. Tel.: þ886 3 2654315; fax: þ886 3 2654351. E-mail address: [email protected] (Y. Kang).

0301-679X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.05.015

journal (PBGJ); the results revealed that the whirl vibration can be induced in PBGJ because the groove number multiplying the rotating speed can be an excitation frequency to disturb the journal leaving its equilibrium position. Lee and his colleagues [7–9] modified Elrod’s method [10] and applied coordinate transformation to analyze three different groove designs in the investigation of the footprints in cavitation distributions within a HGJB. In addition, they investigated the differences between HGJB and PJB to conclude that HGJB is better than PJB on the operating stability and the region of cavitation distributions is also decreased. Rao and Sawicki [11] used the perturbation method and coordinate transformation to solve Elrod’s cavitation equation to study the static load capacity, dynamic stiffness and damping coefficients, stability threshold, and critical whirl ratio to yield that HGJB is better than PJB at a low eccentricity ratio. This study applies the finite difference method associated with Gaussian elimination method to solve the lubricant film Reynolds ¨ equation with the Grumbel boundary conditions. Except Kang et al., the aforementioned researchers all use other various numerical methods. The Gaussian elimination method is faster than an iterative method in solving a set of simultaneous equations. This study will also pay attention to the characteristics of the lubricant flow both in the regular and inverse directions. The aim of this study is to choose and optimize the design parameters of a HGJB to establish the design rules and analysis techniques for the miniature spindle motors with high rotating speed.

2. Finite difference method for solving perturbation of oil film For a hydrodynamic HGJB as shown in Fig. 1, the herringbone grooves with the rectangular cross section having depth hg and

16

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

Q Z ,Q Z

Nomenclature Aij,k

Bij ,Bij

c cd D, R eo ,ep F f ,F f F x ,F y Gij h,h ho ,ho hg Ic K ij ,K ij

L lg, lr m,mc N Ob ,Oj P,P Pa l Plk ,Pk

applied by eight point discrete grid method; the eight points are separated equally on the closed border Gij,l surrounding the chosen grid node Gij , where k¼1–8 and l ¼1–4 the normal and cross damping coefficients and their dimensionless forms Bij ¼ Bij co=Pa LD (i, j ¼x, y) in the Cartesian coordinate, and ði,j ¼ e, fÞ in the perturbation coordinate radial clearance (m) decaying exponent of perturbed whirl diameter and radius of bearing (m) static eccentricity and radial displacement of perturbed whirl (m) friction force due to shear stress within the film (N), and its dimensionless form, F f ¼ F f =P a LD force components due to lubricant film pressure in the x and y directions (N) grid node with the position coordinate (i, j) local film thickness (m), and its dimensionless form, h ¼ h=c static film thickness (m), and its dimensionless form, ho ¼ ho =c groove depth (m) 2 the threshold of instability, Ic ¼ mc lc the normal and cross stiffness coefficients and their dimensionless forms K ij ¼ K ij c=P a LD (i, j ¼x, y) in the Cartesian coordinate, and ði,j ¼ e, fÞ in the perturbation coordinate axial length of bearing (m) groove and ridge width (m) dimensionless mass parameter, m ¼ mco2 =2P a LD; and critical value for m number of herringbone grooves bearing center, and journal (spindle) center the pressure distribution of lubricant film (N/m2), P ¼ P=P a atmospheric pressure (105 N/m2) lubricant film pressure distributions (N/m2); l P k ¼ P lk =P a ; subscript k denoted for o (static), e and f (for perturbation); superscript l denoted for real part by R, and for imaginary part by I

width lg are arranged with bilateral symmetry along the z-axis. The groove angle b is positive when the direction of circumferential flow Qc is the same as shown in this figure, and b is negative when the direction of Qc is inverse. The Reynolds number for this HGJB at the maximum rotational speed is Re ¼

rVD 886  0:002  7500  0:004 ¼ 1476:67, ¼ 0:036 m

which is less than the upper limit of the value: Re ¼ 2300 for laminar flow. With the assumptions of incompressible and laminar flow for lubricant, which is fed into the clearance between the plain journal and the grooved sleeve, and using the following dimensionless parameters: h ¼ h=c, P ¼ P=Pa , y ¼ x=R, z ¼ z=L=2, and t ¼ ot, the governing dimensionless Reynolds equation can be expressed in the yz plane as !   ! ! 3 @P 3 @P @ D 2 @ @h @h h h þ 2l þ ¼L @y @y L @z @z @y @t

ð1Þ

s t W k ,W k

X,Y x, y, z

volumetric flow rate (m3/s) of lubricant, and its  1 dimensionless form Q Z ¼ Q Z P a c3 =12mR , Z ¼ c,s for the circumferential flow and the side leakage characteristic frequency ratio of the journal motion, s ¼ ðcd þ iop Þ=o time (s) unidirectional load capacity (N), and its dimensionless form, W k ¼ W k =Pa LD, k ¼ x,y in the Cartesian coordinate, and k ¼ e, f in the perturbation coordinate dimensionless amplitudes of whirl motion in x and y directions, X ¼ X=c, Y ¼ Y=c Cartesian coordinates of lubricant film

Greek symbols

y,y,z y~

Gij , Gij,k L Oij

a b

g dk

e0 , ep fo , fp l, lc

m mf t o op

dimensionless coordinates, y ¼ x=R, y ¼ y=c, z ¼ z=ðL=2Þ relative angle measured from the line connecting equilibrium positions of journal center and bearing center, y~ ¼ yf0 the virtual closed border and its four segments surrounding the grid node Gij, k ¼1–4 bearing parameter, L ¼ 6mo=Pa ðc=RÞ2 the computational mesh of the node Gij surrounded by Gij,k groove width ratio, a ¼ lg =ðlr þ lg Þ groove angle groove depth ratio, g ¼ hg =c criterion values of convergence, k¼1–3 eccentricity ratios, e0 ¼ e0 =c for static equilibrium, and ep ¼ ep =c for perturbation static attitude angle, and angular displacement of perturbed whirl whirl frequency ratio, l ¼ op =o; and the critical one, lc ¼ op =oc dynamic viscosity of lubricant (N s/m2) the friction coefficient, mf ¼ F f =W dimensionless time, t ¼ ot rotating speed of spindle (rad/s) whirl frequency of perturbed motion (rad/s)

where L ¼ 6mo=P a ðc=RÞ2 is the bearing parameter, l ¼ op =o is the whirl frequency ratio, op is the journal whirl frequency, o is the journal rotating speed, and L and D are the length and diameter of the bearing, respectively. Because of the bilateral symmetry of the herringbone grooves, the calculating domain of Eq. (1), which is only the lower half of the lubricant film as shown in Fig. 1(b), can be numerically solved to obtain its load capacity, side leakage, total friction force, and dynamic characteristics. As the results are multiplied by two the exact complete characteristics of this film are obtained. It is divided into m  n grids as shown in Fig. 2, with m nodes along the lateral axis and n nodes along the vertical axis for numerical computations. The node spans are Dy ¼ 2p=m in the y-axis and Dz ¼ 1=n in the z-axis. Three representative grids A, B and C, located on the interior, the symmetric axis and the periphery of the calculating domain respectively, and the details of their grid structure are shown in Fig. 3, where refers to a grid node and refers to the eight points of neighborhood in the eight-point discrete grid [12]. As shown in Fig. 3, Oij contains node Gij that is

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

17

z journal grooved sleeve

journal

lg c

hg

bearing

groove

ridge

Qc ω

β lg

R

lr

Qs

θ Fig. 1. Hydrodynamic herringbone grooved journal bearing. (a) Cross-section of spindle- bearing system. (b) Expansion graph of herringbone grooves.

z

groove

ridge

B symmetric axis

β

one of edges

point of symmetric axis of grooved sleeve and the edge of one groove as shown in Fig. 3(b), the boundary condition is @P in =@z ¼ 0 so that P i,n þ 1 ¼ P i,n1 and their coefficients for the equation of P i,n are also equal. When a grid node is located at the cross point of the bearing’s periphery and the edge of one groove as shown in Fig. 3(c), its boundary conditions will be P i,1 ¼ 1, i ¼ 12m. The dimensionless rate of net flow passing through this grid can be determined by surface integration of Eq. (1) as follows: "

ZZ Oij

!# !   ! ZZ 3 @P 3 @P @ D 2 @ @h @h h h L þ 2l dy dz ¼ þ dy dz @y @y L @z @z @y @t Oij

ð2aÞ

x=R

A (interior ) C

periphery

Fig. 2. Grid system of calculating domain.

where the surface integral along Oij on the left-hand-side of Eq. (2a) can be transformed into a closed line integral on the basis of Green’s theorem as follows: ! # ZZ ! ! I "  2 3 @P 3 @P D @h @h  h L þ 2l dz ¼ dy dz dy þ h L @z @y @y @t Gij Oij ð2bÞ where Gij ¼

located at the center, which is situated at the coordinates of ði1Þ  ð2pR=mÞ þ ðj1Þ  ðL=2nÞcot b in the lateral direction and ðj1Þ  ðL=2nÞ in the vertical direction measured from the lower left-hand corner as the origin point, which is denoted for each grid with the width being Dy and the length being Dz=sin b. For a typical point within the interior as shown in Fig. 3(a), the approximate values of film thickness for the four terminal points Aij,1 , Aij,4 , Aij,5 , and Aij,8 , are the average of those for their adjacent four grid nodes, and the values for the four inner points Aij,2 , Aij,3 , Aij,6 , and Aij,7 , are the average of three times of the film thickness for the adjacent terminal point and one time of that for the far terminal point. For grid nodes located at the cross

P4

k ¼ 1 Gij,k is the perimeter of Oij and consists of Gij,1 : Aij,1 Aij,4 , Gij,2 : Aij,4 Aij,5 , Gij,3 : Aij,5 Aij,8 , and Gij,4 : Aij,8 Aij,1 , and

the direction along the integrating path is counterclockwise. The discontinuity of lubricant film at the edge between each groove and ridge makes the partial differential term ð@h=@yÞi,j in Eq. (2b) meaningless. The method of eight-point discrete grid is applied to determine the approximation of ð@h=@yÞi,j by the numerical integration of film thickness along the surrounding border to deal with the discontinuity problem of film thickness in differential determination. According to this method, the first term in the left-hand-side of Eq. (2b) can be divided as the summation of four line integrals along the segments from Gij,1 to Gij,4 . However, the projection of Gij,2 on the y-axis is Dz cos b

18

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

edge

Gi- 1, j+1

Gi, j+1

(i - 1, j+1)

Gi+1, j+1

Aij,4 A ij,3

Δz−

1 i − 4 G i, j

(i- 1, j)

1

i− 4

A ij,5 A ij,6

A ij,7 A ij,8

1

z

Gi+1, j- 1 (i+1, j - 1)

(i, j-1)

edge

i+2

Γij,4

θ

Δθ

Gi,n- 1

Gi- 1, n- 1

sin β

(i +1, j)

1

Ωij j− 2

Gi, j- 1

Gi- 1, j- 1

Δ− z

Gi+1, j

(i, j)

Γij,3

(i- 1, j - 1)

Aij,2 Aij,1 Γij,1

Γij,2

1

i− 2

Gi- 1, j

(i+1, j+1)

(i, j+1)

1

j+ 2

Consequently !  2  2 I h 3 3 @P D D  h ðP i,j þ 1 P i,j Þ ðh Þið1=4Þ, j þ ð1=2Þ dy ¼ L @z L Gij i Dy D2 h 3 3  þ ðh Þi þ ð1=4Þ, j þ ð1=2Þ ðPi,j Pi,j1 Þ ðh Þið1=4Þ, jð1=2Þ L 2Dz i Dy 3 ð4Þ þ ðh Þi þ ð1=4Þ, jð1=2Þ 2Dz ! I h 3 3 @P h dz ¼ ðPi þ 1,j Pi,j Þ ðh Þi þ ð1=2Þ, j þ ð1=2Þ @ y Gij i Dz h 3 3 ðP i,j P i1,j Þ ðh Þið1=2Þ, j þ ð1=2Þ þ ðh Þi þ ð1=2Þ, jð1=2Þ 2Dy i Dz 3 ð5Þ þ ðh Þið1=2Þ, jð1=2Þ 2Dy ZZ

Gi+1,n- 1

Oij

edge

Ain,5 Ain,6

Δz− 2 G i- 1, n

1

i− 2

G i, n

(i - 1, n)

Δz− 2

Γin,3

Ain,5 Ain,6 Gi,n- 1

Gi- 1, n- 1 (i - 1, n - 1)

1

n? 2

edge

Gi+1,n

1 1 (Γ ) 2 Ω in 2 in,4 Ain,7 A in,8

Gi+1,n- 1

(i, n -1)

z Δ−

2 sin β

z

(i+1, n - 1)

θ

Δθ

(i- 1, 2)

Gi- 1, 1 (i - 1, 1)

1 (Γ ) 2 i1,2

1

i− 2

Ai1,4 Ai1, 3 Γi1, 3 Gi, 1

1+ 2

Ai1,2 A i1,1 1 (Γ ) 1 2 i1,4 2 ? i1 Gi+1,1

1 1 i − 4 (i, 1) i + 4

Δθ

1

i+2

(i+1,1)

Δz− 2 sinβ z

θ

3

h

Gij,1

j þ ð1=2Þ

j þ ð1=2Þ ðhÞið1=2Þ, j þ ð1=2Þ jð1=2Þ ðhÞið1=2Þ, jð1=2Þ

Gij,3

3

h

! h 3 @P dy ¼ ðP i,j P i,j1 Þ ðh Þið1=4Þ, @z

i Dy j þ ð1=2Þ 2Dz

3 jð1=2Þ þ ðh Þi þ ð1=4Þ, jð1=2Þ

i Dz

ð6Þ

2

ð7Þ

ð8Þ

at ridges, and h ¼ h0 þ g þ ep est cos y~ þ e0 fp est sin y~

y

ð9Þ

φ0 θ

∼ θ

W Fx

O Oj

φ

de dt

x

εc

e

ε

O

e

ω

i Dy 2Dz

ð3bÞ

Fy

φ

dφ dt

O

ð3aÞ Z

i þ ð1=2Þ

ðhÞ9ið1=2Þ dz

jð1=2Þ

P ¼ P0 þ ep est Pe þ e0 fp est Pf

The trapezoidal rule is applied to approximate the line integrals of Eq. (3) as h 3 3 @P dy ¼ ðPi,j þ 1 P i,j Þ ðh Þið1=4Þ, j þ ð1=2Þ þ ðh Þi þ ð1=4Þ, @z

Z

The eight points from Aij,1 to Aij,8 of grid node Gij do only for the determinations of approximate film thickness as mentioned above and do nothing for the determinations of film pressure. Due to a small perturbation, the journal center leaves its equilibrium position ðeo , fo Þ as shown in Fig. 4 and whirls with radial and tangential components Reðep est Þ and Reðeo fp est Þ, which induce pressure and thickness perturbations on the film. In the first-order approximation, the dimensionless pressure perturbation can be expressed as

and that of Gij,4 will be Dz cos b; then, the summation of the line integrals along Gij,2 and Gij,4 equals to zero. Thus, the integration yields ! ! ! I Z Z 3 @P 3 @P 3 @P dy ¼ dy þ dy h h h ð3Þ @z @z @z Gij Gij,1 Gij,3

!

ið1=2Þ

@h dy dz ¼ @y

h ¼ h0 þ ep est cos y~ þ e0 fp est sin y~

periphery

Fig. 3. Details of computational grid structure. (a) Details of A. (b) Details of B. (c) Details of C.

Z

i þ ð1=2Þ

where P0 ¼ P 0 =P a is the dimensionless static pressure, P e ¼ P e =P a and P f ¼ P f =Pa are the radial and tangential dimensionless perturbation pressures respectively, fp is the angular displacement of perturbation, and s ¼ ðcd þ iop Þ=o is the characteristic frequency ratio of journal whirl. In a similar manner, the dimensionless thickness of film perturbation can be expressed as

Gi+1, 2 (i+1, 2)

(i, 2)

1

z Δ− 2

edge

G i, 2

Gi- 1, 2

jð1=2Þ

Z

þðhÞi þ ð1=2Þ,

(i +1, n)

(i, n)

1 i− 4

1 (Γ ) 2 in,2

i+

1 i+ 4

1 2

j þ ð1=2Þ

h ¼ ðhÞi þ ð1=2Þ,

symmetric axis

Ain,7 A in,8

Z

@h dy dz ¼ @y

ε

ε φc

φ

Fig. 4. Whirling motion of lubricant film within a hydrodynamic bearing: (a) the static equilibrium position and (b) the perturbation coordinates of journal center.

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

at grooves, where y~ ¼ yf0 is the relative angle measured from the line connecting equilibrium positions of journal center and bearing center, h0 ¼ h0 =c is the dimensionless static film thickness with h0 ¼ hw for ridges and h0 þ g ¼ hw for grooves, and g ¼ hg =c is the groove depth ratio. Substituting Eqs. (7)–(9) into Eq. (2b) and neglecting perturbation terms higher than the first order gives ! # ZZ ! I "  2 3 @P 0 3 @P 0 D @h dz ¼ dy þ hw  hw L w dy dz L @z @y @y Gij Oij

ð10Þ

o

o

Oij

ð11Þ of a dynamic part that is obtained from perturbation terms with ep est , and !   !# I "  2 3 @P f D D 2 2 @P 0 ~  hw hw sin y 3 dy L L @z @z Gij ! I 3 @P f 2 @P 0 þ hw þ 3hw sin y~ dz @y @y Gij ZZ   c L cos y~ þ 2 d sin y~ dy dz ¼

Wx ¼

Wx ¼ 2 P a LD

Z 0

1

Z 2p

P 0 cos y~ R dy dz ¼ 2

0

n X m X

ðP0 cos y~ Þij RDy Dz

j¼1i¼1

ð14aÞ in the lateral direction of static load, which should be zero, and Z 1 Z 2p n X m X Wy ¼2 Wy ¼ P 0 sin y~ R dy dz ¼ 2 ðP 0 sin y~ Þij RDy Dz P a LD 0 0 j¼1i¼1

in the negative direction of static load. For solving Eq. (13) iteratively, an initial guess includes both distributions of static pressure ðP 0 Þij and attitude angle fo. Iterations are terminated when three conditions as described below are satisfied:   W    new old old Max9ðP k Þnew   r d2 , and Dfo ¼ 9fo fo 9 r d3 ij ðP k Þij 9 r d1 , W  ð15Þ where superscripts new and old denote the newest result and the last one of iteration, and d1, d2, and d3 are obtained by a negotiation between accuracy and computing time. The solution process expressed in flow chart is shown in Fig. 5. In Eq. (10), the first term in the left-hand-side is the volumetric flow rate of the grid perpendicular to dy. For the value of (D/L) being unit, the dimensionless side leakage Q s in the z-axis is obtained by Z 1 Z 2p 3 @P 0 12mR Q s 12mR R dy dz Qs ¼ ¼ 2 h0 @z P a c3 P a c3 0 0 n X m i 3 24mR X 1 h ðP 0 Þi,j þ ð1=2Þ ðP 0 Þi,jð1=2Þ R Dy Dz ðh0 Þij ¼ 3 Dz Pa c j ¼ 1 i ¼ 1

o

Oij

ZZ þi

ð16Þ

 o  L 2 p sin y~ dy dz

ð12Þ

o

Oij

of another dynamic part that is obtained from perturbation terms with e0 fp est , on both sides of this equation. The central difference method can be used for numerical integration of Eqs. (10)–(12), which are discretized in the same form as follows:

The second term in the left-hand-side of Eq. (10) is the volumetric flow rate of the grid perpendicular to dz, and the right-hand-side is that referring to the rotating shaft and the oil viscosity along the y-axis. So, the dimensionless circumferential flow rate Q c in the y-axis of the lubricant film is obtained by Qc ¼

12mR Q c 12mR ¼ 2 Pa c3 P a c3

Z

1 0

Z 2p 0

3

Lh0 h0

! @P 0 R dy dz @y

n X m i

3 24mR X 1 h ¼ Lðh0 Þij ðh0 Þij ðP 0 Þi þ ð1=2Þ, j ðP 0 Þið1=2Þ,j R Dy Dz 3 Dy Pa c j ¼ 1 i ¼ 1

Ak,ij ðPk Þi1,j þ Bk,ij ðP k Þi,j þ C k,ij ðPk Þi þ 1,j þDk,ij ðP k Þi,j1 þ Ek,ij ðP k Þi,j þ 1 ¼ F k,ij

ð17Þ

ð13Þ

where P k represents P 0 and the complex forms: Pe ¼ ReðPe Þ þ R

determined from ðP 0 Þij by

ð14bÞ

of the static part that is obtained from the constant terms, and !   !# I "  2 3 @P e D D 2 2 @P 0  hw hw cos y~ 3 dy L L @z @z Gij ! I 3 @P e 2 @P 0 dz þ hw þ 3hw cos y~ @y @y Gij ZZ ZZ  o  c Lðsin y~ þ2 d cos y~ Þdy dz þ i L 2 p cos y~ dy dz ¼ Oij

19

I

R

I

i ImðP e Þ ¼ P e þ iPe and P f ¼ ReðP f Þ þ i ImðPf Þ ¼ P f þ iP f ; F k,ij is the complex form of the non-homogeneous terms of Eqs. (10)–(12), and k¼ 0, e, or f. Further, Eq. (13) can be expanded into real and R

I

R

I

imaginary parts and five equations for P 0 , P e , Pe , P f , and Pf are obtained. The details about coefficients Ak,ij to F k,ij of five equations, which are expressed by Eq. (13), are shown in Appendix A. Eq. (13) represents m  n coupled equations that describe the flow rate passing through the m  n grids.

Because the total friction force of the film is the surface integral of the shear stress along Oij due to the rotation of journal, the dimensionless total friction force F f of the lubricant film in the y-axis is obtained by ! Z 1 Z 2p Ff h0 @P 0 L R dy dz ¼ mf W Ff ¼ ¼2 þ 2cLP a 4 @y 12h0 0 0 ( ) n X m i X ðh0 Þij 1 h L ¼2 ðP 0 Þi þ ð1=2Þ,j ðP 0 Þið1=2Þ,j þ R Dy Dz 4 Dy 12ðh0 Þij j¼1i¼1 ð18Þ where mf ¼ F=W is the friction coefficient.

3. Static analyses 4. Dynamic stiffness and damping coefficients For Eq. (13) of ðP0 Þij , the coefficients and non-homogeneous term F 0,ij are functions only of dimensionless static film thickness. The components of the dimensionless load capacity can be

The perturbation load capacity W e induced by the infinitesimal perturbation term ep est Pe , is the surface integral along the

20

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

START

END

Input geometric parameters: L , R , c , N , , , Input physical parameters: , , , , Pa Input convergence parameters: grids , relaxation factor, iteration threshold of iteration error initial guess of attitude angle o and initial guess of static pressure P o

Calculate film thickness and coefficients of finite difference Reynolds equation (18) Solve static part of finite difference Reynolds equation using Gauss -Siedel method to determine nodal static pressures

(P ) = (P ) +

No

0

α [( P0 ) − ( P0 ) new

c

Compute dynamic coefficients K ij , B

ij

Yes Check convergence on dynamic pressure

0

( Pε ) = ( Pε ) + α [ ( Pε ) − ( Pε ) ] ( Pφ ) = ( Pφ ) + α [ ( Pφ ) − ( Pφ ) ] new

new

No

new

old

]

Check convergence on static pressure

Compute Wx , new and

new

No new

0

−5

/

≤ 10

Δ

≤ 10

W W o

old

old

old

old

Solve dynamic part of finite difference Reynolds equation using Gauss-Siedel method to determine nodal dynamic pressures P R , P R , P I , P I

Wy

( 0)

(P )

c

old

Yes

(φ0 )

Compute critical mass parameter and critical whirl ratio m ,

new

old

new

0

(P )

Output maximum pressure, load capacity, attitude angle, side leakage, stability threshold, critical whirl ratio

Yes

−5

Output static pressures

Fig. 5. Solution scheme.

Ob

lubricant film of this perturbation and can be subdivided into radial and tangential parts as Z ðW e ÞR ¼ 2

1

Z

P e R cos y~ dy dz

1 0

Z 2p

Ob P e R sin y~ dy dz

ð19bÞ

ðW e ÞR ¼

ðWÞR K ee c Bee cop þi ¼ K ee þ ilBee ¼ P a LD Pa LDep P a LD

ð20aÞ

ðW e ÞT ¼

K fe c B fe c o p ðWÞT þi ¼ K fe þ ilBfe ¼ Pa LDep P a LD P a LD

ð20bÞ

Similarly, the perturbation load capacity W f induced by the infinitesimal perturbation term e0 fp est P f is the surface integral along the lubricant film of this perturbation and can be subdivided as Z 1 Z 2p ðW f ÞR ¼ 2 Pf R cos y~ dy dz ð21aÞ 0

ðW f ÞT ¼ 2

0

0

1 Z 2p 0

Pf R sin y~ dy dz

ð21bÞ

enter

y

e0

(e0 , 0 )

0

0 p

(e0 p , ep)

m

0

However, this perturbation load capacity is also a linear combination of stiffness coefficients and damping coefficients and can be non-dimensionalized as

Z

ng c Beari

ð19aÞ

0

0

ðW e ÞT ¼ 2

Z 2p

of orbit whirl center al j ourn Oj

Oj

x

p

Fig. 6. Model of rotor–hydrodynamic-bearing system.

Table 1 Dimensionless design parameters of HGJB. Bearing length to diameter ratio, L/D Groove angle, b Groove number, N Groove width ratio, a Groove depth ratio, g Eccentricity ratio, e Bearing parameter, L

1 20–701,  20 to  701 8 1/4, 1/3, 1/2 0.5, 1.0, 1.5, 2.0 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 188.4956, 75.39822

This perturbation load capacity is also a linear combination of stiffness coefficients and damping coefficients and can be nondimensionalized as ðW f ÞR ¼

K ef c Bef cop ðWÞR þi ¼ K ef þilBef ¼ P a LDeo fp Pa LD Pa LD

ð22aÞ

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

ðW f ÞT ¼

K ff c Bff cop ðWÞT þi ¼ K ff þ ilBff ¼ P a LD P a LDeo fp P a LD

21

and they can be obtained directly by integrating the dimensionless real and imaginary parts of perturbation pressures Pe and Pf of Eqs. (19) and (21) as

ð22bÞ

where Bij ¼ Bij co=Pa LD and K ij ¼ K ij c=Pa LD (i,j ¼ e, f). Hence, Eqs. (20) and (22) are the dimensionless stiffness and damping coefficients under the condition of any whirl frequency ratio l,

K ee ¼ 2

Z 0

1

Z 2p 0

n X m X

R P e ðy,zÞR cos y~ dy dz ¼ 2

R ðP e cos y~ Þij R Dy Dz

j¼1i¼1

1000

6 numerical results of Lee et al [9] present results experimental results of Hirs [1]

W

T (sec)

4

6 6× 25

6 8× 25

25

12

12

8 ×1 2

8× 12

0.8

4

0.6

64

0.4 ε

×6

0.2

64

0 0.0

8

100

2

grid number (m×n) Fig. 7. Evaluation of the validity and time spent for present method. (a) Result comparison for load capacity. (b) Computing time for static analyses and the determination of dynamic coefficients.

12

90 =1/2 =1/3 =1/4

W

8 6

c=6 m : • 3 m: ×

PJB3, PJB6

70 o

10

PJB3, PJB6

4

=1/2 =1/3 =1/4

50 c=6 m : • 3 m: ×

2 0 0.0

0.2

0.4

0.6

30 0.0

0.8

= 20° = 40° = 60° = 70°

W

8 c=6 m : • 3 m: ×

= 20° = 40° = 60° = 70°

PJB3, PJB6

50 c=6 m : • 3 m: ×

0.2

0.4

0.6

30 0.0

0.8

12

0.2

0.4

0.6

= 0.5 = 1.0 = 2.0

PJB3, PJB6 = 0.5 = 1.0 = 2.0 c=6 m : • 3 m: ×

c=6 m : • 3 m:×

70 o

8

W

0.8

90

10

6

0.8

70

2 0 0.0

0.6

PJB3, PJB6

o

10

4

0.4

90

12

6

0.2

PJB3, PJB6

4

50

2 0 0.0

0.2

0.4

0.6

0.8

30 0.0

0.2

0.4

0.6

0.8

Fig. 8. Static dimensionless load capacity (W) and attitude angle (fo) versus eccentricity ratio (e) for PJB and various design parameters of HGJB with various clearances. (a) b ¼ 301, g ¼ 0.5, with various a. (b) a ¼ 1/4, g ¼ 0.5, with various b. (c) a ¼ 1/4, b ¼ 301 with various g.

22

K fe ¼ 2

K ef ¼ 2

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

Z

Z 2p

1 0

0

Z

K ff ¼ 2

lBee ¼ 2

0

0

Z

1

Z

1

lBef ¼ 2

lBff ¼ 2

Z 0

Z 0

Z 0

equations are "

K xx

K xy

K yx

K yy

#

" ¼

sin fo

#2 K ee 4 cos fo K fe

cos fo sin fo

#2 Bee sin fo 4 cos fo Bfe

cos fo

3" cos fo 5 sin fo K ff þ W x K ef W y

sin fo

sin fo

#T

cos fo

R ðP f cos y~ Þij R Dy Dz

Z 2p

ð24Þ "

R

ðP f sin y~ Þij R Dy Dz

Bxx

Bxy

Byx

Byy

#

" ¼

j¼1i¼1

3" Bef cos fo 5 sin fo Bff

sin fo cos fo

#T

ð25Þ I

P e ðy,zÞR cos y~ dy dz ¼ 2

Z 2p

Z 2p 0

1 Z 2p 0

n X m X

R

0

1

n X m X

P f ðy,zÞR sin y~ dy dz ¼ 2

0

1

R

ðP e sin y~ Þij R Dy Dz

j¼1i¼1

0

0

lBfe ¼ 2

R P f ðy,zÞR cos y~ dy dz ¼ 2

Z 2p

0

n X m X j¼1i¼1

Z 2p

1

R

P e ðy,zÞR sin y~ dy dz ¼ 2

n X m X

I

ðP e cos y~ Þij R Dy Dz

j¼1i¼1 n X m X

I P e ðy,zÞR sin y~ dy dz ¼ 2

I ðP e sin y~ Þij R Dy Dz

j¼1i¼1

I P f ðy,zÞR cos y~ dy dz ¼ 2

n X m X

I ðP f cos y~ Þij R Dy Dz

j¼1i¼1

I P f ðy,zÞR sin y~ dy dz ¼ 2

n X m X

I ðPf sin y~ Þij R Dy Dz

j¼1i¼1

ð23Þ The stiffness and damping coefficients obtained from Eq. (23) are expressed in e–f coordinate system. To make the follow-up comparison more clear, we need to transform the coordinate system of the results from e–f to x–y. The transformation

5. Stability analysis For a balanced rigid rotor, supported horizontally by two identical HGJB on both ends, the shaft with a concentrated mass m as the rotor at its center is shown in Fig. 6. When the rotor is disturbed, the center of rotor will deviate from its equilibrium position. This rotor center will whirl with a complex frequency cd þ iop , and the whirl motion is dependent on the dynamic characteristics of the rotor-bearing system. If cd o0, the rotor center will eventually whirl back to its equilibrium position; if cd 40, the rotor center whirls farther away from its equilibrium position as time increases and will cause damage to bearing or journal when contact occurs. When cd ¼0, which is the critical condition between stability and instability, the track of rotor center will be a circle, so called limit circle. With a small disturbance, the dimensionless perturbation equation of the journal with whirl center at Oj in a referenced x–y coordinate

4

1200

7x10

600

6x10

4

5x10

c=6 m : • 3 m:×

Qc

Qs

800

4

=1/2 =1/3 =1/4

1000

3x10

PJB3

4 4

PJB6

0.2

0.4

1x10

0.6

6.0x104

Qc

Qs

c=6 m : • 3 m: ×

500

0.6

0.0 0.0

0.8

1.0x105

2.0

8.0x104

(×10 )

= 0.5 = 1.0 = 2.0

0.6

0.8

= 0.5 = 1.0 = 2.0

c=6 m : • 3 m:×

PJB3

PJB3

2.0x104

PJB6

0.4

0.4

4.0x104

c=6 m : • 3 m:×

0.2

PJB6

0.2

6.0x104

0.5 0.0 0.0

= 20° = 40° = 60° = 70°

Qc

1.5

0.8

2.0x104 PJB6

0.4

0.6

PJB3

PJB3

0.2

4.0x10

c=6 m : • 3 m:×

0.4

4

2.5

1.0

0.2

8.0x104 = 20° = 40° = 60° = 70°

1000

Qs

PJB6

0.0

0.8

1500

0 0.0

c=6 m : • 3 m:×

2x10

200 0 0.0

=1/2 =1/3 =1/4

4

PJB3

400

4

4x10

0.6

0.8

0.0 0.0

PJB6

0.2

0.4

0.6

0.8

Fig. 9. Dimensionless side leakage (Q s ) and total circumferential flow rate (Q c ) versus eccentricity ratio (e) for PJB and various design parameters of HGJB with various clearances. (a) b ¼301, g ¼ 0.5, with various a. (b) a ¼1/4, g ¼0.5, with various b. (c) a ¼ 1/4, b ¼301 with various g.

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

system can be described by #" # "  " € # " Bxx Bxy K xx x_ m 0 x þ þ € y_ Byx Byy K yx 0 m y

#" # K xy x K yy

y

¼

 0

ð26aÞ

0

where m ¼ mco2 =ð2Pa LDÞ is the dimensionless mass parameter. For the special case of cd ¼0, we have s ¼ iop =oc ¼ ilc , x ¼ Xeilc t and y ¼ Yeilc t . Substituting x and y into Eq. (26a) gives " #" #  K xx þilc Bxx Ic K xy þ ilc Bxy 0 X ¼ ð26bÞ 0 K yx þ ilc Byx K yy þ ilc Byy Ic Y

23

where Ic can be directly determined by Eq. (27) after K ij and Bij have been calculated. Then, lc can be obtained by substituting Ic into Eq. (28). Finally, the dimensionless parameter of critical mass 2 can be obtained by dividing Ic by l2c , written as mc ¼ Ic =lc . The stability condition for whether rotor whirls or not is also dependent on the dimensionless mass parameter, m. The rotor is stable when m is less than mc ; on the other hand, the rotor is unstable if m is larger than mc .

6. Results and discussions where X and Y are the whirl amplitudes in x and y directions, 2 respectively, Ic ¼ mc lc is the threshold of instability and represents the impedance of the self-excited vibration. The determinant of Eq. (26b) must be zero for non-trivial solutions of X and Y. Thus, setting the determinant of Eq. (26b) equal to zero both in imaginary and real parts, the critical dynamic stabilities Ic and l2c of this rotor-bearing system can be obtained as follows: Ic ¼

l2c ¼

K xx Byy þ K yy Bxx K xy Byx K yx Bxy

ð27Þ

Bxx þ Byy ðIc K xx ÞðIc K yy ÞK xy K yx

ð28Þ

Bxx Byy Bxy Byx

Based on the above analyses, a computational program is built up to calculate the static and dynamic characteristics of a HGJB with the dimensionless design parameters as shown in Table 1. These data were chosen to represent a miniature hydrodynamic bearing usually used in a hard disk drive system. A detailed comparison among the present results and the experimental data of Hirs [1] and the numerical results of Lee et al. [9] is shown in Fig. 7(a). The present results match very well with their results. The major difference between HGJB and PJB is the influence of grooves. In order to master the design techniques of a HGJB, the influences of design parameters on the static and dynamic characteristics are investigated in great detail. According to the

3

800

10 =1/2 =1/3 =1/4

600

2

c=6 m : • 3 m: ×

10

400

c=6 m : • 3 m: ×

f

Ff

=1/2 =1/3 =1/4

PJB3

10

PJB6

200

PJB3, PJB6

0 0.0

0.2

0.8

0.2

0.4

0.6

0.8

10

= 20° = 40° = 60° = 70°

600

= 20° = 40° = 60° = 70°

PJB3 2

10

c=6 m : • 3 m: ×

c=6 m : • 3 m: ×

μf

Ff

0.6

3

800

400

0.4

1 0.0

10 PJB6

200

PJB3, PJB6

0 0.0

0.2

0.4

0.6

1 0.0

0.8

0.6

0.8

3

= 0.5 = 1.0 = 2.0

600

= 0.5 = 1.0 = 2.0

PJB3

10

c=6 m : • 3 m: ×

2

c=6 m : • 3 m: ×

μf

Ff

0.4

10

800

400

0.2

10 200 0 0.0

PJB6

0.2

0.4

PJB3, PJB6

0.6

0.8

1 0.0

0.2

0.4

0.6

0.8

Fig. 10. Dimensionless friction force (F f ) and friction coefficient (mf) versus eccentricity ratio (e) for PJB and various design parameters of HGJB with various clearances. (a) b ¼ 301, g ¼ 0.5, with various a. (b) a ¼ 1/4, g ¼ 0.5, with various b. (c) a ¼ 1/4, b ¼ 301 with various g.

24

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

detailed comparison among the computed results, the optimum design of a HGJB for the most stable and maximum load capacity can be obtained. As shown in Table 1, the complete design parameters are taken into account for the analyses of both static and dynamic characteristics of a HGJB. The minus sign of groove angle b signifies the flow at inverse direction. The time spent for various grid meshes is calculated as shown in Fig. 7(b). Because of the application of matrix operating techniques, the time spent in calculating is very little. For the need of accuracy, the grid mesh of 128  256 is applied in both static and dynamic cases and the time spent is 380 s on average. The clearance has no effect on both the dimensionless load capacity W and the attitude angle fo. W increases with the increase of e, and that of PJB is a little higher than those of HGJB at the beginning, then the difference between them increases rapidly with the increase of e as shown in Fig. 8. However, fo

4x108

3x106

5x10 3x10

PJB3

2x10

6

2x10

PJB6

B xx

2x108

HGJB3 HGJB6

PJB3

4x10

3x108

K xx

decreases with the increase in e. In order to find out the influence of groove width ratio a on the W and fo of a HGJB, groove angle b ¼301 and groove depth ratio g ¼0.5 were chosen in the present study. The simulation results indicate that the W of PJB is greater than all of HGJB after e ¼0.2 as shown in Fig. 8(a). The fo of HGJB with various a are very close and slightly less than that of PJB. Choosing a ¼1/4 and g ¼0.5, the simulation results indicate that the W of HGJB with various b are very near and all less than that of PJB as shown in Fig. 8(b). The fo of PJB and HGJB with various b are all very close for various e. Choosing a ¼1/4 and b ¼301, the simulation results indicate that the sequence of W is g ¼ 0:5 4 g ¼ 1:0 4 g ¼ 2:0 for HGJB as shown in Fig. 8(c). The sequence of fo is also the same. Both the dimensionless side leakage Q s and circumferential flow rate Q c increase with the increase in e, and those of PJB are always less than those of HGJB as shown in Fig. 9. The results show very clear that both Q s and Q c are inversely proportional to

1x10 0 0.1 HGJB3 HGJB6

1x108

0.2

PJB3

1x106

0.3

PJB6 PJB6

0 0.0

0.2

0.4

0.6

0 0.0

0.8

0.2

0.4

0.6

0.8

2.0x106

1.0x109

8.0x10

HGJB3 HGJB6

8.0x108

PJB3

4.0x108

1.0x106

PJB3

PJB3

2.0x10

PJB6

0.0

5.0x10

2.0x108 PJB6

0.0 0.0

0.2

0.4

0.6

0.1 HGJB3 HGJB6

5

0.0 0.0

0.8

0.0

0.2

0.2

0.3

0.4

0.4

0.6

0.8

2.0x106 2.5x10

PJB6

-2.0x108

1.5x106

2.0x10 1.5x10

PJB3

PJB6

1.0x10

-4.0x108

B yx

K yx

6.0x10 4.0x10

8

B xy

K xy

6.0x10

1.5x106

PJB3

1.0x106

-6.0x108 5.0x105

HGJB3 HGJB6

-8.0x108 -1.0x109 0.0

0.2

5.0x10 0.0 -5.0x10

0.1

HGJB3 HGJB6

0.4

0.6

0.0 0.0

0.8

6.0x108

0.2

0.2

0.3

PJB3

0.4

0.6

0.8

3x106 2.5x10

HGJB3 HGJB6

2.0x10 8 1.5x10

PJB3

1.0x10

2x10 B yy

K yy

4.0x10

6

PJB3

5.0x10

2.0x108

0.0 0.05

0.10

0.15

PJB6

HGJB3 HGJB6

0.0 0.0

0.2

PJB3

1x106

0.20

PJB6

0.4

0.6

0.8

PJB6

0 0.0

0.2

0.4

0.6

0.8

Fig. 11. Dynamic stiffness coefficients (K ij ) and damping coefficients (Bij ) versus eccentricity ratio (e) for PJB and HGJB with various clearances, and a ¼1/2, b ¼ 301, g ¼ 0.5.

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

the cube of clearance. The results of Q s are very close for HGJB with various a as shown in Fig. 9(a). However, the sequence of Q c is a ¼ 1=4 4 a ¼ 1=34 a ¼ 1=2. The sequence of Q s is b ¼ 701 4 b ¼ 601 4 b ¼ 4014 b ¼ 201 for HGJB with various b as shown in Fig. 9(b). However, the results of Q c are very close for various e. The sequence of Q s is g ¼ 2:0 4 g ¼ 1:0 4 g ¼ 0:5 for HGJB as shown in Fig. 9(c); the sequence of Q c is the same. The dimensionless total friction force F f increases with the increase in e, and that of PJB is always greater than those of HGJB as shown in Fig. 10. The values of F f with clearance c ¼3 mm are four times that of F f with clearance c¼6 mm but the friction coefficient mf has the same value. However, mf has an abrupt drop from a high value near 150–20 when e goes from 0.01 to 0.1, and goes down to near unit when e ¼0.8. The sequence of F f is a ¼ 1=2 4 a ¼ 1=3 4 a ¼ 1=4 for HGJB as shown in Fig. 10(a). The results of F f are very close for HGJB with various b as shown in Fig. 10(b). The sequence of F f is g ¼ 2:0 4 g ¼ 1:0 4 g ¼ 0:5 for HGJB as shown in Fig. 10(c). Both the dynamic stiffness coefficients K ij excluding K yx and damping coefficients Bij increase with the increase in e, and those of PJB are greater than those of HGJB when e 40.4 as shown in Fig. 11. PJB3 and PJB6 are denoted for PJB with clearances 3 mm and 6 mm. HGJB3 and HGJB6 are denoted similarly and both with a ¼1/2, b ¼301, and g ¼0.5. The sequences of K xy , Bxx , Bxy and Byy

are very clear, which are PJB34HGJB34PJB64HGJB6. In the case of K yx , their values are all negative and they decrease with the increase in e; the sequence is HGJB64PJB64HGJB3 4PJB3. The clearance also has no effect on both the stability threshold mc and the critical whirl frequency ratio lc. mc of HGJB decreases from a large value near 90 down to about 13, however that of PJB is nearly unchanged at about 12 in comparison with HGJB before e ¼ 0:2, and then they go inversely at near 13 all the way to e ¼ 0:8 as shown in Fig. 12. Except two in HGJB with various b and two in HGJB with various g, lc of HGJB and that of PJB decrease at first and then increase rapidly with the increase of e. The lc of those four in HGJB go down for large eccentricity. The minus groove angle has great influence on the performances of the stability threshold mc and the friction coefficient mf for HGJB with a ¼ 1=2 and g ¼ 0:5 as shown in Fig. 13. The friction coefficient for HGJB with b ¼ 201 is about five times greater than that with b ¼ 401, which is over double of the rest when e ¼0.01, and then they all come close to each other and near unit. The stability thresholds for HGJB with minus groove angle, excluding b ¼ 701, are all negative which means the systems are unstable. The performance of PJB keeps unchanged regardless whether the spindle rotates in regular or inverse direction. In the other conditions, the minus groove angle for HGJB has little effect.

0.60

90 α =1/2 α =1/3 α =1/4

c=6μ m : • 3μ m : ×

50

instability

0.4 ε

0.6

0.45 0.0

0.8

β = 20° β = 40° β = 60° β = 70°

50

10 0.0

0.55

c=6μ m : • 3μ m : ×

instability

0.4 ε β = 20° β = 40° β = 60° β = 70°

0.6

0.8

PJB3, PJB6

stability

×

0.50 PJB3, PJB6

instability

stability 0.2

0.4 ε

0.6

0.45 0.0

0.8

200

0.2

0.4 ε

0.6

0.8

0.70 stability

γ = 0.5 γ = 1.0 γ = 2.0

150

0.65 0.60

c=6μ m : • 3μ m : ×

100

λc

mc

c=6μ m : • 3μ m :

λc

mc

70

PJB3, PJB6

stability

0.2

0.60

90

30

c=6μ m : • 3μ m : ×

instability

stability 0.2

α =1/2 α =1/3 α =1/4

0.50

PJB3, PJB6

10 0.0

0.55

λc

mc

70

30

25

c=6μ m : • 3μ m :

γ = 0.5 γ = 1.0 γ = 2.0

×

0.55 50

instability

0.50

PJB3, PJB6

instability

PJB3, PJB6

0 0.0

0.2

stability

0.4 ε

0.6

0.8

0.45 0.0

0.2

0.4 ε

0.6

0.8

Fig. 12. Stability threshold (mc ) and critical whirl speed ratio (lc) versus eccentricity ratio (e) for PJB and various design parameters of HGJB with various clearances. (a) b ¼ 301, g ¼ 0.5, with various a. (b) a ¼ 1/4, g ¼ 0.5, with various b. (c) a ¼ 1/4, b ¼ 301 with various g.

26

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

15

90

W

70 0

= -20° = -40° = -60° = -70°

10

PJB

5

0

= -20° = -40° = -60° = -70°

50

0.0

0.2

0.4

0.6

30 0.0

0.8

0.2

1.4

300 = -20° = -40° = -60° = -70°

0.4

0.6

0.8

= -20° = -40° = -60° = -70°

(×10 4)

Qc

Qs

200

PJB

1.0

100

PJB 0

PJB 0.0

0.2

0.4

0.6

0.8 0.0

0.8

0.2

0.4

0.6

0.8

4

200

10

Ff

150

PJB

= -20° = -40° = -60° = -70°

3

10 f

= -20° = -40° = -60° = -70°

2

10

100 10

PJB 50

0.0

0.2

0.4

0.6

1 0.0

0.8

20

0.2

0.8

= -20° = -40° stability = -60° = -70°

10 0.55 = -20° = -40° = -60° = -70°

0 -10

stability 0.0

0.2

0.4

0.6

c

mc

0.6

0.60

PJB

instability

-125

0.4

0.50

PJB 0.8

0.45 0.0

0.2

instability 0.4

0.6

0.8

Fig. 13. Characteristics of various bearing design versus eccentricity ratio (e) with clearance 6 mm for PJB and HGJB with a ¼1/2, g ¼0.5 and minus groove angle b. (a) Static ¯ s) and circumferential flow (Q ¯ c). (c) Dimensionless friction force (F¯f) and friction coefficient (mf). ¯ ) and attitude angle (f0). (b) Dimensionless side leakage (Q load capacity (W (d) Dynamic stability threshold (m ¯ c) and whirl frequency ratio (lc).

7. Conclusions The conclusions of this study are as follows. (1) The static load capacity W, the side leakage Q s , the circumferential flow rate Q c , and the total friction force F f for all kinds of bearings increase with the increase in eccentricity ratio e. (2) For the stability threshold mc , those of HGJB are much larger than PJB at low eccentricity ratio, but they are nearly the same beyond the threshold. (3) For a HGJB with journal rotating along the inverse direction, the stability threshold mc is negative, which means that the rotor-bearing system is unstable.

(4) The bearing clearance c has no effect on the static load capacity W, the attitude angle fo, the stability threshold mc , and the critical whirl ratio lc. (5) From the above analyses, the optimum design parameters can be obtained. The hydrodynamic bearing with herringbone grooved sleeve having a ¼1/2, b ¼401, and g ¼ 1.0 will be the best choice for L/D ¼1 and light loading applications irrespectively of the value of clearance.

Acknowledgment The authors greatly appreciate the support of this work by Grants NSC 92-2212-E-033-016 from the National Science Council

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

and MEA 98-EC-17-A-01-02-0704 from the Ministry of Economic Affairs, Taiwan, ROC.

Appendix A. The coefficients of Eq. (13) for P 0 , Pe , and P f The details about coefficients Ak,ij to F k,ij of five equations for R I R I P 0 , Pe , P e , Pf , and P f expressed by Eq. (13) are given as follows. 1. Coefficients of P 0 : !  2  2 h I i 1 3 @P 0 D D  hw ðP0 Þi,j ðP0 Þi,j1 dy ¼  L L @z Dz Gij  2 h h 3 i i 3 Dy D  ðh0 Þið1=4Þ, jð1=2Þ þ ðh0 Þi þ ð1=4Þ, jð1=2Þ ðP 0 Þi,j þ 1 ðP 0 Þi,j þ 2 L i Dy 3 1 h 3 ðh0 Þi þ ð1=4Þ, j þ ð1=2Þ þ ðh0 Þi þ ð1=4Þ, j þ ð1=2Þ  Dz 2 ! I h i 1 3 @P 0 hw dz ¼  ðP 0 Þi,j ðP 0 Þi1,j @ y Dy Gij h 3 i Dz 3  ðh0 Þið1=2Þ,jð1=2Þ þðh0 Þið1=2Þ,j þ ð1=2Þ 2 h i 1 h 3 ðh0 Þi þ ð1=2Þ,jð1=2Þ þ ðP 0 Þi þ 1,j ðP0 Þi,j Dy i Dz 3 þ ðh0 Þi þ ð1=2Þ,j þ ð1=2Þ 2 ! ZZ Z j þ ð1=2Þ Z i þ ð1=2Þ @hw @h0 L dy dz ¼ L dy dz @y @y Oij jð1=2Þ ið1=2Þ Dz h ðh0 Þi þ ð1=2Þ, j þ ð1=2Þ ðh0 Þið1=2Þ, j þ ð1=2Þ ¼L 2 þðh0 Þi þ ð1=2Þ, jð1=2Þ ðh0 Þið1=2Þ, jð1=2Þ 

ðA1bÞ

Z

j þ ð1=2Þ

Z

jð1=2Þ

ZZ

2L Oij

ðA2dÞ

i þ ð1=2Þ

! ~ sin y dy dz

ið1=2Þ

Dz h

ðcos y~ Þ

cd

o

¼L

cos y~ dy dz ¼ 2L

cd

o

ðA1cÞ

ZZ i

cd

o

Z

Z

j þ ð1=2Þ

jð1=2Þ

i þ ð1=2Þ

ið1=2Þ, jð1=2Þ

ðsin y~ Þ

ðA2eÞ !

cos y~ dy dz

ið1=2Þ

h Dz ðsin y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðsin y~ Þið1=2Þ,

þ ðsin y~ Þ

j þ ð1=2Þ

ið1=2Þ, jð1=2Þ 

Z

ðA2fÞ !

Z

i þ ð1=2Þ op op j þ ð1=2Þ cos y~ dy dz ¼ i2L cos y~ dy dz o o jð1=2Þ Oij ið1=2Þ op h ¼ iL Dz ðsin y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðsin y~ Þið1=2Þ, j þ ð1=2Þ o

2L

þ ðsin y~ Þi þ ð1=2Þ,

3

jð1=2Þ ðsin

y~ Þið1=2Þ, jð1=2Þ 

ðA2gÞ

Substituting Eqs. (A2a)–(A2g) into Eq. (11) and rearranging it gives Eq. (13) with the following coefficients for real and imaginary parts of P e : R

1. for real part Pe : ARe,ij ¼ A0,ij , F Re,ij

j þ ð1=2Þ þðh0 Þi þ ð1=2Þ, jð1=2Þ ðh0 Þið1=2Þ, j þ ð1=2Þ

 2  2 h i 1 2 D @P0 D dy ¼ 3 3 hw cos y~ ðP 0 Þi,j ðP0 Þi,j1 L L @z Dz Gij h 2 i Dy 2  ðh0 cos y~ Þið1=4Þ, jð1=2Þ þ ðh0 cos y~ Þi þ ð1=4Þ, jð1=2Þ 2  2 h i 1 D ðP0 Þi,j þ 1 ðP 0 Þi,j þ3 L Dz h 2 i Dy 2  ðh0 cos y~ Þið1=4Þ, j þ ð1=2Þ þ ðh0 cos y~ Þi þ ð1=4Þ, j þ ð1=2Þ 2

L sin y~ dy dz ¼ L

jð1=2Þ

jð1=2Þ

~ i þ ð1=2Þ, j þ ð1=2Þ ðcos y Þið1=2Þ, j þ ð1=2Þ 2 þ ðcos y~ Þi þ ð1=2Þ, jð1=2Þ ðcos y~ Þið1=2Þ, jð1=2Þ 

BRe,ij ¼ B0,ij , C Re,ij ¼ C 0,ij , DRe,ij ¼ D0,ij , ERe,ij ¼ E0,ij R R ¼ Ge,ij Se,ij ðP 0 Þi1,j T Re,ij ðP 0 Þi,j U Re,ij ðP 0 Þi þ 1,j V Re,ij ðP 0 Þi,j1 Z Re,ij ðP 0 Þi,j þ 1

h GRe,ij ¼ L ðcos y~ Þi þ ð1=2Þ,

ðh0 Þið1=2Þ, jð1=2Þ Dz

I

i Dz 2 þ ð3h0 cos y~ Þið1=2Þ, j þ ð1=2Þ 2 h i 1 h 2 þ ðP0 Þi þ 1,j ðP 0 Þi,j ð3h0 cos y~ Þi þ ð1=2Þ, Dy i Dz 2 þ ð3h0 cos y~ Þi þ ð1=2Þ, j þ ð1=2Þ 2

¼L

3

2. Coefficients of P e :  2  2 h I i 1 3 @P e D D dy ¼   hw ðP e Þi,j ðPe Þi,j1 L L @z Dz Gij h 3 i Dy 3  ðh0 Þið1=4Þ, jð1=2Þ þ ðh0 Þi þ ð1=4Þ, jð1=2Þ 2  2 h i 1 D ðP e Þi,j þ 1 ðPe Þi,j þ L Dz h 3 i Dy 3  ðh0 Þið1=4Þ, j þ ð1=2Þ þðh0 Þi þ ð1=4Þ, j þ ð1=2Þ 2

h i 1 h 2 2 @P 0 3hw cos y~ dz ¼  ðP 0 Þi,j ðP 0 Þi1,j ð3h0 cos y~ Þið1=2Þ, @ y Dy Gij

Oij

C 0,ij ¼ ðh0 Þi þ ð1=2Þ, j þ ð1=2Þ þ ðh0 Þi þ ð1=2Þ, jð1=2Þ  2  2 h i 3 3 D Dy D0,ij ¼ ðh0 Þið1=4Þ, jð1=2Þ þ ðh0 Þi þ ð1=4Þ, jð1=2Þ L Dz  2  2 h i 3 3 D Dy ðh0 Þið1=4Þ, j þ ð1=2Þ þ ðh0 Þi þ ð1=4Þ, j þ ð1=2Þ E0,ij ¼ L Dz B0,ij ¼ ðA0,ij þC 0,ij þ D0,ij þ E0,ij Þ F 0,ij ¼ L½ðh0 Þi þ ð1=2Þ,

ðA2cÞ

I

ZZ

A0,ij ¼ ðh0 Þið1=2Þ, j þ ð1=2Þ þ ðh0 Þið1=2Þ, jð1=2Þ 3

h i 1 h 3 @P e dz ¼  ðP e Þi,j ðP e Þi1,j ðh Þ @y Dy 0 ið1=2Þ, jð1=2Þ Gij i Dz h i 1 3 þ ðP e Þi þ 1,j ðPe Þi,j þ ðh0 Þið1=2Þ, j þ ð1=2Þ 2 Dy h 3 i Dz 3  ðh0 Þi þ ð1=2Þ, jð1=2Þ þ ðh0 Þi þ ð1=2Þ, j þ ð1=2Þ 2 3

hw

ðA1aÞ

Substituting Eqs. (A1a)–(A1c) into Eq. (10) and rearranging it gives Eq. (13) with the following coefficients: 3

I

27

j þ ð1=2Þ þ ðcos

y~ Þi þ ð1=2Þ, jð1=2Þ

i ðcos y~ Þið1=2Þ, j þ ð1=2Þ ðcos y~ Þið1=2Þ, jð1=2Þ Dy c h þ L d ðsin y~ Þi þ ð1=2Þ, j þ ð1=2Þ þ ðsin y~ Þi þ ð1=2Þ, jð1=2Þ o i ðsin y~ Þið1=2Þ, j þ ð1=2Þ ðsin y~ Þið1=2Þ, jð1=2Þ Dy h 2 i 2 SRe,ij ¼ 3 ðh0 cos y~ Þið1=2Þ, j þ ð1=2Þ þ ðh0 cos y~ Þið1=2Þ, jð1=2Þ h 2 i 2 U Re,ij ¼ 3 ðh0 cos y~ Þi þ ð1=2Þ, j þ ð1=2Þ þ ðh0 cos y~ Þi þ ð1=2Þ, jð1=2Þ ðA2aÞ

 2  2 h 2 D Dy Z Re,ij ¼ 3 ðh0 cos y~ Þið1=4Þ, L Dz  2  2 h 2 D Dy ðh0 cos y~ Þið1=4Þ, Z Re,ij ¼ 3 L Dz

i

2 j þ ð1=2Þ þ ðh0

cos y~ Þi þ ð1=4Þ,

j þ ð1=2Þ

2 j þ ð1=2Þ þ ðh0

cos y~ Þi þ ð1=4Þ,

j þ ð1=2Þ

i

T Re,ij ¼ ðSRe,ij þ U Re,ij þ V Re,ij þ Z Re,ij Þ I

2. for imaginary part Pe : ðA2bÞ

AIe,ij ¼ A0,ij ,

BIe,ij ¼ B0,ij ,

C Ie,ij ¼ C 0,ij ,

DIe,ij ¼ D0,ij ,

EIe,ij ¼ E0,ij

28

S.-K. Chen et al. / Tribology International 55 (2012) 15–28

op h F Ie,ij ¼ L ðsin y~ Þi þ ð1=2Þ,

j þ ð1=2Þ þðsin

o

ðsin y~ Þið1=2Þ,

j þ ð1=2Þ ðsin

F Rf,ij ¼ GRf,ij SRf,ij ðP0 Þi1,j T Rf,ij ðP 0 Þi,j U Rf,ij ðP0 Þi þ 1,j

y~ Þi þ ð1=2Þ, jð1=2Þ

V Rf,ij ðP 0 Þi,j1 Z Rf,ij ðP 0 Þi,j þ 1

y~ Þið1=2Þ, jð1=2Þ Dy

3. Coefficients of Pf :  2  2 h I i 1 3 @P f D D dy ¼   hw ðP f Þi,j ðP f Þi,j1 L L @z Dz Gij  2 h 3 i 3 Dy D þ  ðh0 Þið1=4Þ, jð1=2Þ þ ðh0 Þi þ ð1=4Þ, jð1=2Þ L 2 h i 1 h 3 i Dy 3 ðh0 Þið1=4Þ, j þ ð1=2Þ þ ðh0 Þi þ ð1=4Þ, j þ ð1=2Þ  ðPf Þi,j þ 1 ðP f Þi,j Dz 2 ðA3aÞ  2  2 h i 1 2 D @P0 D dy ¼ 3 3 hw sin y~ ðP 0 Þi,j ðP 0 Þi,j1 L L @z Dz Gij h 2 i Dy 2  ðh0 sin y~ Þið1=4Þ, jð1=2Þ þ ðh0 sin y~ Þi þ ð1=4Þ, jð1=2Þ 2  2 h i 1 D ðP 0 Þi,j þ 1 ðP0 Þi,j þ3 L Dz h 2 i Dy 2 ðA3bÞ  ðh0 sin y~ Þið1=4Þ, j þ ð1=2Þ þ ðh0 sin y~ Þi þ ð1=4Þ, j þ ð1=2Þ 2

I

I

Gij

h i 1 h 2 @P 0 siny~ dz ¼  ðP 0 Þi,j ðP 0 Þi1,j ð3h0 siny~ Þið1=2Þ, @y Dy 2

þ ð3h0 siny~ Þið1=2Þ, j þ ð1=2Þ h

þ ðP 0 Þi þ 1,j ðP 0 Þi,j

i 1 h

Dy

ðA3cÞ

Z

2 y~ Þi þ ð1=2Þ,jð1=2Þ þ ð3h0 sin

Z

j þ ð1=2Þ jð1=2Þ

Oij

y~ Þi þ ð1=2Þ,j þ ð1=2Þ

i þ ð1=2Þ

ZZ

2L

2

!

ið1=2Þ

Dz h

Oij

cd

o

¼ L

c sin y~ dy dz ¼ 2L d

o

2L Oij

jð1=2Þ

Z

i þ ð1=2Þ

sin y~ dy dz

ið1=2Þ

h

jð1=2Þ ðcos

op op sin y~ dy dz ¼ i2L o o

¼ iL

j þ ð1=2Þ

ðA3eÞ !

Dz ðcos y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðcos y~ Þið1=2Þ, j þ ð1=2Þ

þ ðcos y~ Þi þ ð1=2Þ, ZZ i

Z

o

cd

Z

y~ Þið1=2Þ, jð1=2Þ 

j þ ð1=2Þ

jð1=2Þ

Z

i þ ð1=2Þ

ðA3fÞ ! sin y~ dy dz

ið1=2Þ

op h Dz ðcos y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðcos y~ Þið1=2Þ, j þ ð1=2Þ o

þ ðcos y~ Þi þ ð1=2Þ,

jð1=2Þ ðcos

y~ Þið1=2Þ, jð1=2Þ 

ðA3gÞ

Substituting Eqs. (A3a)–(A3g) into Eq. (12) and rearranging it gives Eq. (13) with the following coefficients for real and imaginary parts of P f : R

1. for real part Pf : ARf,ij ¼ A0,ij ,

BRf,ij ¼ B0,ij ,

BIf,ij ¼ B0,ij , C If,ij ¼ C 0,ij , DIf,ij ¼ D0,ij , EIf,ij ¼ E0,ij op h F If,ij ¼ L ðcos y~ Þi þ ð1=2Þ, j þ ð1=2Þ þ ðcos y~ Þi þ ð1=2Þ, jð1=2Þ

o

ðcos y~ Þið1=2Þ, j þ ð1=2Þ ðcos y~ Þið1=2Þ,

jð1=2Þ Dy

i Dz

cos y~ dy dz

ðsin y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðsin y~ Þið1=2Þ, j þ ð1=2Þ 2 þ ðsin y~ Þi þ ð1=2Þ, jð1=2Þ ðsin y~ Þið1=2Þ, jð1=2Þ 

¼L

T Rf,ij ¼ ðSRf,ij þ U Rf,ij þ V Rf,ij þ Z Rf,ij Þ

References

2

L cos y~ dy dz ¼ L

2

þ ðh0 sin y~ Þi þ ð14Þ, j þ ð1=2Þ 

jð1=2Þ

ðA3dÞ ZZ

i ðsin y~ Þið1=2Þ, j þ ð1=2Þ ðsin y~ Þið1=2Þ, jð1=2Þ Dy c h L d ðcos y~ Þi þ ð1=2Þ, j þ ð1=2Þ ðcos y~ Þið1=2Þ, j þ ð1=2Þ o i þ ðcos y~ Þi þ ð1=2Þ, jð1=2Þ ðcos y~ Þið1=2Þ, jð1=2Þ Dy h 2 i 2 SRf,ij ¼ 3 ðh0 sin y~ Þið1=2Þ, j þ ð1=2Þ þðh0 sin y~ Þið1=2Þ, jð1=2Þ h 2 i 2 U Rf,ij ¼ 3 ðh0 sin y~ Þi þ ð1=2Þ, j þ ð1=2Þ þ ðh0 sin y~ Þi þ ð1=2Þ, jð1=2Þ  2  2 h 2 D Dy V Rf,ij ¼ 3 ðh0 sin y~ Þið1=4Þ, jð1=2Þ L Dz i 2 þ ðh0 sin y~ Þi þ ð1=4Þ, jð1=2Þ  2  2 h 2 D Dy ðh0 sin y~ Þið1=4Þ, j þ ð1=2Þ Z Rf,ij ¼ 3 L Dz

AIf,ij ¼ A0,ij ,

i Dz

2 ð3h0 sin

y~ Þi þ ð1=2Þ, jð1=2Þ

2. for imaginary part Pf :

hw

2 3hw

j þ ð1=2Þ þ ðsin

I

3

Gij

I

h i 1 h 3 @P f dz ¼  ðPf Þi,j ðP f Þi1,j ðh Þ @y Dy 0 ið1=2Þ, jð1=2Þ i Dz h i 1 3 þ ðPf Þi þ 1,j ðPf Þi,j þ ðh0 Þið1=2Þ, j þ ð1=2Þ 2 Dy h 3 i Dz 3  ðh0 Þi þ ð1=2Þ, jð1=2Þ þ ðh0 Þi þ ð1=2Þ, j þ ð1=2Þ 2

h GRf,ij ¼ L ðsin y~ Þi þ ð1=2Þ,

C Rf,ij ¼ C 0,ij ,

DRf,ij ¼ D0,ij ,

ERf,ij ¼ E0,ij

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