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Lubrication performance of short journal bearings considering the effects of surface roughness and magnetic field
Tribology International 61 (2013) 169–175
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Lubrication performance of short journal bearings considering the effects of surface roughness and magnetic field Tze-Chi Hsu a,n, Jing-Hong Chen a, Hsin-Lu Chiang b, Tsu-Liang Chou b a b
Department of Mechanical Engineering, Yuan-Ze Unversity, Chung-Li 320, Taiwan, ROC Department of Mechanical Engineering, Nanya Institute of Technology, PO Box 267, Chung-Li 320, Taiwan, ROC
a r t i c l e i n f o
abstract
Article history: Received 22 May 2012 Received in revised form 16 November 2012 Accepted 11 December 2012 Available online 11 January 2013
The primary objective of this study was to investigate the performance of ferrofluids under the combined influence of surface roughness and a magnetic field generated by a concentric finite wire. The distribution of pressure was obtained and used to calculate the characteristics of the bearing. The results of this study indicate that the combined influence affect the distribution of film pressure, which enhances loading capacity and reduces the modified friction coefficient. The effect of longitudinal roughness was particularly significant and we expect these findings could further enhance the design of short journal bearing within a magnetic field. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Short journal bearing Ferrofluid Christensen stochastic roughness theory Concentric finite wire magnetic field
1. Introduction Ferrofluids are a non-Newtonian fluid with the strong magnetism of solid magnetic materials and the fluidity of liquids. A number of studies [1–6] have shown that ferrofluids have exceptional lubricating characteristics capable of improving the load capacity of bearings, reducing wear, and lengthening the operational life. Ferrofluids have been studied in various practical applications, including the finishing of optical instruments, fluid clutches, sealants, and in automotive and civil damping [7–10]. Osman et al. [11–15] investigated the influence of ferrofluid lubricants on the operational characteristics of journal bearings. In addition to deriving generalized ferrofluid Reynolds equations, they investigated the effects of ferrofluids using specifically designed magnetic field models. Their results have indicated that increasing the power law index of ferrofluids and the intensity of magnetic fields can enhance the loading capacity and attitude angle of the bearing and decrease the modified friction coefficient of the bearing in the case of high eccentricity. However, the model of the applied magnetic field must be selected with care to prevent the occurrence of side leakage due to excessive film pressure. Huang et al. [16] verified through experimentation that Fe3O4 ferrofluid could be used to improve the static loading
n Correspondence to: Department of Mechanical Engineering, Yuan-Ze Unversity 135 Yuan-Tung Road, Chung-Li 320, Taiwan, ROC. Tel.: þ 886 3 4638800x2458; fax: þ 886 3 4558031. E-mail addresses: [email protected], [email protected] (T.-C. Hsu).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.12.016
capacity and friction characteristics of bearings under the influence of an external magnetic field. These studies all hypothesized that the surfaces of the bearings are completely smooth. Zhu [17] showed that manufacturing processes often produce microscopic surface features displaying transverse, longitudinal, and isotropic surface roughness. Christensen and Tonder [18,19] developed stochastic Reynolds equations to describe average pressure, addressing transverse and longitudinal roughness, and determined the influence of roughness on the performance of bearings. Chiang et al. [21–23] derived a generalized stochastic Reynolds equation based on Christensen’s stochastic model and the Stokes microcontinuum theory [24]. Through numerical simulation, the performances of lubrication in bearings under the combined influence of couple stress and surface roughness was determined to characterize the dynamic squeeze film. Hsu et al. [25] applied Christensen’s stochastic roughness theory between two parallel circular disks, their results of which demonstrate that surface roughness could improve squeeze film behavior. Magnetic particles are often trapped in the valleys associated with surface roughness; however, previous studies have tended to focus on the impact of surface roughness, without considering the influence of magnetic fields. The objective of the present study was to investigate the combined effects of surface roughness as well as magnetic fields. This study addressed short journal bearings with a length-todiameter ratio of r0.25. A magnetic field was created using a concentric metal wire with finite length to investigate the operational characteristics of a short journal bearing lubricated with non-Newtonian power law ferrofluid under the influences of
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Mg n Pm p, P R Xm x, y, z Z
Namenclature C C e fMX fMZ hm hm0 Hm h hm, H L m0
One half of the total range of the random film thickness variable Radial clearance Eccentricity, e ¼ eC Magnetic forces in x-direction (circumferential direction) Magnetic forces in z-direction (axial direction) Magnetic field intensity Characteristic value of magnetic field intensity Dimensionless magnetic field intensity, HM ¼ hM =hM0 Thickness of lubricant film Nominal smooth part of the film thickness, H¼hm/ C ¼1þ ecosy Length of the bearing Viscosity constant
am e d y l L
m0 o
Magnetization of ferrofluid Power-law index Induced magnetic pressure n þ 1 Mean film pressure, P ¼ ðp==m0 on ÞðÞ C=R Radius of the journal Susceptibility of ferrofluid Rectangular coordinates Dimensionless coordinate in the z-direction, Z¼ z/L Magnetic field coefficient, am ¼ ðh2M0 m0 X m =m0 on ÞðC=RÞn þ 1 Eccentricity ratio, e ¼ e=C Random part of film geometry Circumfer ential coordinate, x ¼Ry Length-to-diameter, l ¼ L=2R Roughness parameter, L ¼c/C Permeability of free space of air, m0 ¼4p 10 7 AT/m Angular speed
n
Fig. 1 depicts the configuration of a short bearing with a journal radius R rotating at angular speed o, with lubrication of the interior bearings provided by ferrofluid. A finite metal wire is located in the center of the bearing, through which electric current I induces a magnetic field. Based on the Navier–Stokes equation, the magnetic force was treated as a external body force and the modified Reynolds equation was then obtained as [14] nþ2
f MX
þ
@ nþ2 h f MZ @z
ð1Þ
where n is the power-low index, p is local film pressure, and m0 is the viscosity constant. The fMX and fMZ are the components of magnetic force in the circumferential and axial directions, respectively, written as follows:
2. Theoretical analysis
@ h @x n
!
nþ2
@h @ h ¼ 6m0 ðRoÞ þ @x @x n
bearing surface roughness. It is hoped that our establishment of theoretical model could be used as a reference for the future design of bearings.
f MX ¼ m0 X M hM
@hM @x
ð2Þ
f MZ ¼ m0 X M hM
@hM @z
ð3Þ
here m0 is permeability of free space of air, m0 ¼4p 10 7 AT/m. XM is the susceptibility of ferrofluid. In the concentric field model with a finite wire the intensity of the magnetic field hM provides a magnetic field represented by [14] I L=2 þ z L=2z þ sintan1 ð4Þ sintan1 hM ðzÞ ¼ 4pR R R
! @p @ n þ 2 @p þ h @x @z @z
where I is the strength of the current passing through the wire.
cutaway view y θ
y
x
L
Concentric finite wire
z e
h
R
U
I I
x
h hm 1
A. Transverse Roughness
= 1 + 2 2 = 12 + 22
2
Fig. 1. Configuration of a journal bearing.
B. Longitudinal Roughness
T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
According to Christensen’s theorem using expected values of Eq. (1), the stochastic modified Reynolds equation for the journal bearing with rough surface can be written as: " ! #
nþ2 @ h @p @ n þ 2 @p E E h þ @x @x @z @z n " # ! nþ2 @EðhÞ @ h @ n þ 2 þ E E h f MZ ð5Þ f MX þ ¼ 6m0 ðRoÞn @x @x @z n where the expectancy operator E( ) is defined by Z 1 EðUÞ ¼ ðUÞf ðdÞdd
The non-dimensional modified Reynolds equation for longitudinal roughness can then be expressed as
@ @P n @ @P @H G2 ðH, L,nÞ ¼ 6n þ 2 G1 ðH, L,nÞ @y @y @Z @Z @y 4 l
@ @HM nam @ @HM G ð15Þ þ am G1 ðH, L,nÞHM ð H, L ,n ÞH þ M 2 2 @y @y @Z 4l @Z where hm C
ð16Þ
Z ¼ z=L
ð17Þ
L 2R
ð18Þ
y ¼ x=R
ð19Þ
H¼ ð6Þ
171
1
and f(d) is the probability density distribution for the stochastic variables. As most engineering rough surfaces are Gaussian in nature, one can choose for simplicity, instead of using a Gaussian distribution function for integration, a polynomial form as following: ( 2 35 ðc2 d Þ3 if c r d r c f ðdÞ ¼ 32c7 ð7Þ 0 elsewhere where c is the half total range of random film thickness variable and, the function terminates at c¼ 73s; where s is the standard deviation. In this study, local film geometry of the lubricant is treated as a stationary, ergodic, stochastic process with zero mean, written as h ¼ hm ðx,zÞ þ dðx,z, xÞ
ð8Þ
where hm represents the nominal smooth part of the film geometry depending upon the coordinates x and z, while d(x,z,x) is the part due to the surface asperities measured from the nominal level and is regarded as a randomly varying quantity of zero mean. Using the circumferential coordinate, the film thickness hm can also be expressed as hm ¼ C ð1 þ e cosyÞ
ð9Þ
where C is the radial clearance, and e is eccentricity ratio, e ¼ e=C. The surface feature of one-dimensional longitudinal roughness is assumed to have the form of long narrow ridges and valleys running in the direction of rotation. Thus, the thickness of the lubricating film listed in Eq. (8) can be expressed as a function of the form h ¼ hm ðx,zÞ þ dðz, xÞ
ð10Þ
whereupon Eq. (5) for longitudinal roughness can be reduced to 2 3 " ! # nþ2 @ h @p @ 4 1 @p 5 E þ @x @x @z E 1=hn þ 2 @z n 0 1 " # ! nþ2 @E ð h Þ @ h @ 1 f A þ E ¼ 6m0 ðRoÞn f MX þ @ @x @x @z E 1=hn þ 2 MZ n
l¼
G1 ðH, L,nÞ ¼ Hn þ 2 þ
G2 ðH, L,nÞ ¼
1 H
nþ2
1 n 2 2 H L n þ 3n þ2 18
þ
L2 ½n2 þ 5n þ 6 18 Hn þ 4
ð20Þ
ð21Þ
and the intensity of the dimensionless magnetic field of Eq. (4) can be written as
HM ðZÞ ¼ sin tan1 l þ2lZ þsin tan1 l2lZ ð22Þ where HM ¼ hM =hM0
ð23Þ
and hM0 ¼ I=4pR
ð24Þ
One-dimensional transverse roughness is assumed to have the form of long narrow ridges and valleys running in the direction perpendicular to rotation. Therefore, the lubricant film thickness can be expressed as a function of the form: h ¼ hm ðx,zÞ þ dðx, xÞ Eq. (15) for transverse roughness can be derived as
@ @P n @ @P G1 ðH, L,nÞ G2 ðH, L,nÞ þ 2 @y @y @Z 4l @Z
@ @ @HM ¼ 6n G:3 ðH, L,nÞ þ am G2 ðH, L,nÞHM @y @y @y
nam @ @HM G1 ðH, L,nÞHM þ 2 @Z 4l @Z
ð25Þ
ð26Þ
where G3 ðH, L,nÞ ¼ H
18H2 þ L2 ½n2 þ5n þ6 18H2 þ L2 ½n2 þ 7n þ 12
ð27Þ
where p is mean film pressure. To simplify the analysis, the magnetic field coefficient can be defines as n þ 1 2 h m X C am ¼ M0 0 n M ð12Þ R m0 o
Because l{ 1 for the approximation of a short journal bearing, the circumferential variation in pressure can be neglected in favor of axial variations. Therefore, Eqs. (15) and (26) are reduced to
@ @P 2 @H G2 ðH, L,nÞ ¼ 24l @Z @Z @y
@ @HM ðZÞ G2 ðH, L,nÞHM ðZ Þ Longitudinal ð28Þ þ am @Z @Z
and dimensionless mean film pressure P and surface roughness parameter L, written as n þ 1 p C ð13Þ P¼ m0 on R
@ @P @ 2 G1 ðH, L,nÞ ¼ 24l G3 ðH, L,nÞ @Z @Z @y
@ @HM ðZÞ G1 ðH, L,nÞHM ðZ Þ Transvers þ am @Z @Z
ð11Þ
L¼
c C
ð29Þ
The boundary conditions for the film pressure are as follows: ð14Þ
P¼0
at
Z ¼ 71=2
ð30Þ
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T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
dP ¼0 dZ
at
Z¼0
0.55
ð31Þ
0.5
applying the boundary conditions (30) and (31), the non-dimensional mean film pressure can be obtained by integrating Eqs. (28) and (29) as 2
P ¼ 12 l e sin y
1 1 am þ Z2 K ðZ Þ 2 G2 ðH, L,nÞ 4
0.45 0.4
Longtudinal
0.35
ð32Þ 2
P ¼ 12l
@ 1 1 am G3 ðH, L,nÞ Z2 K ðZ Þ þ 2 @y G1 ðH, L,nÞ 4
0.3 Transverse
P 0.25
ð33Þ
0.2
where ð34Þ
0.15
By integrating the non-dimensional mean film pressure acting on the journal bearing, the dimensionless load components parallel and perpendicular to the line of centre W0 and Wp/2 can be acquired by Z p 1 2 sin y W 0 ¼ 2l e 0 G2 ðH, L,nÞ Z pZ 1 2 cos y dyam K 1 ðZÞcos ydZdy Longitudinal ð35Þ
0.1
KðZÞ ¼ ðHM ðZÞÞ2 ðHM ðZÞ9Z ¼ 0:5 Þ2
0
2
W 0 ¼ 2l
Z p @
0
G3 ðH, L,nÞ
@y Z pZ
0
1 2
cos y dyam
0
2
Wp=2 ¼ 2l e þ am
Z p 0
Z pZ 0
1 2
1 G1 ðH, L,nÞ
K 1 ðZÞcos ydZdy
Transverse
ð36Þ
1 sin2 ydy G2 ðH, L,nÞ Longitudinal
ð37Þ
0
2
Wp=2 ¼ 2l
Z p @ 0
sin ydy þ am
G3 ðH, L,nÞ
@y Z pZ 0
1 2
1 G1 ðH, L,nÞ
K 1 ðZÞsin ydZdy
Transverse
ð38Þ
0
Thereafter, the dimensionless load capacity W and attitude angle j can be evaluated from above, as follows: W ¼ ½W 0 2 þ W p=2 2 1=2
ð39Þ
j ¼ tan1 ½W p=2 =W 0
ð40Þ
The shear stress on the moving surface in the direction of rotation can also be calculated, such that the modified friction coefficient f R/C is determined by
2 f R=C ¼
f R=C ¼
R 12 R 2p 0
0
1 Hn
0
60
90 θ
120
150
180
The objective of this study was to investigate the influence of ferrofluids on the lubrication performance of short journal bearing systems under the combined effects of surface roughness and a magnetic field model generated by a finite concentric wire. To verify the accuracy of the derived theory and the reliability of the numerical analysis, we compare our results with those of Hsu et al. [20] and Osman et al. [12]. Hsu et al. [20] focused on the influence of surface roughness in journal bearing systems. Under a magnetic force of 0, our numerical analysis of the smooth case is similar to that obtained by Chiang. Furthermore, the increase in the magnetic field contributed to an increase in lubricant pressure, as shown in Fig. 2. On the other hand, Osman et al. [12] focused on the influence of magnetism on bearing characteristics without considering the effects of surface roughness. Fig. 3 presents a comparison of the current simulations and those of Osman (adopting the same assumptions), in which similar results were obtained. The impact of surface roughness was also observed in the current model, in which longitudinal roughness facilitated an increase in Pmax, whereas transverse roughness had the opposite effect. Although the viscosity of ferrofluids changes only slightly in the Schliomis model, a number of studies [11–15] have shown that viscosity can be affected by differences in the weight ratio of
W n h ion R 1 R 2p 2 2 02 0 H1n 1þ g 12l Z 2 14 @@y @@y G3 ðH, L,nÞ G1 ðH,1L,nÞ dydZ
ð1 þ ecos yÞn þ 1 2n
30
Fig. 2. Comparison between current simulations with Hsu et al. [20] (Z¼ 0, n¼ 1, e ¼ 0.6, l ¼ 0.25, L ¼ 0).
h in 2 1 þ g ð12Þ l e Z 2 14 @@y sin y G2 ðH,1L,nÞ dy dZ
W
where
g¼
0
3. Results and discussion
0
K 1 ðZÞsinydZdy
0.05
ð43Þ
Longitudinal
Transverse
ð41Þ
ð42Þ
ferromagnetic particles within the fluid. Therefore, the impact of various power law indices on dimensionless pressure distribution under the influence of surface roughness is shown in Fig. 4. Clearly, with an increase in the power law index, using longitudinal roughness as an example, shear thickening fluid (n¼1.5) has the highest
T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
0.65
2.1 Osman, 2001
1.95
0.6
Smooth, Λ = 0
1.8
Longitudinal, Λ = 0.3
0.55
Transverse, Λ = 0.3
1.65
0.5
1.5
0.45
1.35
0.4
1.2
0.35
1.05
P
P
173
0.3
0.9
0.25
0.75
αm = 0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 αm = 0.5,Smooth,Λ = 0 Longitudinal,Λ = 0.3 Transverse, Λ = 0.3
0.2
0.6 0.45
0.15
0.3
0.1
0.15
0.05
0
0
0
30
60
90
θ
120
150
180
210
Fig. 3. Comparison between current simulations with Osman et al. [12] (Z¼ 0, n¼ 1, e ¼0.5, l ¼ 1, am ¼ 0).
1.1
n = 0.7, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.5, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3
1 0.9 0.8 0.7
P
0.6 0.5 0.4 0.3 0.2 0.1 0 0
30
60
90 θ
120
150
180
Fig. 4. Relationship between dimensionless pressure P and y under the influence of various power law indices and surface roughness (Z ¼0, e ¼ 0.6, l ¼0.25, am ¼ 0.5).
Pmax, followed by Newtonian fluid (n¼1.0) and shear thinning fluid (n¼0.7). Fig. 5 shows the relationship between dimensionless pressure P and Z at y ¼1501 under different magnetic field coefficient. It is clear that increasing the coefficient of the magnetic field derives higher dimensionless pressure in which, Pmax increases considerably. In addition, the same trend is observed that the surface roughness has a significant influence on the distribution of dimensionless pressure P. With am ¼0.5 as an example, longitudinal roughness increased Pmax compared to a smooth bearing surface whereas transverse roughness slightly decreased Pmax.
-0.5
-0.4
-0.3
-0.2
-0.1
0 Z
0.1
0.2
0.3
0.4
0.5
Fig. 5. Relationship between dimensionless pressure P and y under various magnetic field effects and surface roughness (y ¼ 1501, n ¼1, e ¼0.6, l ¼ 0.25).
Table 1 illustrates the combined effects of surface roughness and a magnetic field according to variations in Pmax. When the magnetic field coefficient equaled zero, the maximum dimensionless pressure increased to 17.8%, in conjunction with an increase in longitudinal roughness from L ¼0 to L ¼0.3. In contrast, an increase in transverse roughness slightly decreased the maximum dimensionless pressure, compared to the smooth case. The current study assumed that the short journal bearings had a length-to-diameter ratio equal to 0.25; therefore, circumferential variations in pressure were neglected in favor of those in the axial direction. As a result, the longitudinal grooves generated stronger hydrodynamic lift, resulting in higher maximum dimensionless pressure; however, as the length-to-diameter ratio increased, the dominance of surface roughness patterns on dimensionless pressure changed from longitudinal to transverse. Research is currently underway to investigate this phenomenon with regard to long journal bearings. The influence of a magnetic field on maximum dimensionless pressure is presented in Table 1. Because the surface was assumed to be smooth, the maximum dimensionless pressure increased to 6.7% when the magnetic field coefficient was increased from 0 to 0.5. Finally, combining the influence of surface roughness and magnetic field resulted in a maximum dimensionless pressure of 28.4%, in the case of L ¼0.3 and am ¼0.5. It is clear that with an increase in the magnetic field, the influence of surface roughness is far more pronounced than that of a magnetic field and the coupled effects become more significant. To ensure strong hydrodynamic lift within the journal bearing and prevent side leakage caused by excessive hydrodynamic pressure, an appropriate balance must be attained between these two factors. Fig. 6 exhibits the relationship between dimensionless load W and eccentricity ratio e under various magnetic field coefficients. Clearly, dimensionless load is directly proportional to eccentricity ratio. The increase of dimensionless load due to the increase of magnetic field coefficient is more apparent in the low eccentricity ratio. Moreover, the surface roughness pattern also has an effect on dimensionless load. Taking am ¼0.5 as an example, longitudinal roughness can increase dimensionless load, in comparison with
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T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
Table 1 Pmax as a function of surface roughness at various am (Z ¼0, n ¼1, e ¼ 0.6, l ¼ 0.25). Surface roughness
Magnetic field coefficient am ¼0
Longitudinal L ¼ 0.3 L ¼ 0.1 Smooth L¼0 Transverse L ¼ 0.3 L ¼ 0.1
Pmax 0.597 0.557 0.507 0.492 0.475
Pmax, L P max, am ¼ 0, L ¼ 0 P max, am ¼ 0, L ¼ 0
Pmax, L P max, am ¼ 0, L ¼ 0 P max, am ¼ 0, L ¼ 0
100%(%) Magnetic field coefficient am ¼0.2 Pmax 0.615 0.571 0.518 0.504 0.486
17.8 9.9 0 3.0 6.3
100%(%) Magnetic field coefficient am ¼ 0.5 Pmax 0.651 0.601 0.541 0.528 0.51
21.3 12.6 2.2 0.6 4.1
Pmax, L P max, am ¼ 0, L P max, am ¼ 0, L ¼ 0
¼ 0
100%(%)
28.4 18.5 6.7 4.1 0.6
90
0.45 Smooth, 0.4
Longitudinal, Λ = 0.3
85
Transverse, Λ = 0.3
80
0.35
75
0.3 70 ϕ
W
0.25 65
0.2
n = 0.7,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.5, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3
60
0.15 55
αm = 0.5
0.1
50
αm = 0 0.05
45
0
40
0.1
0.2
0.3
0.4
0.5
0.6
0.1
ε Fig. 6. Relationship between dimensionless load and eccentricity ratio under various magnetic field effects and surface roughness (n¼ 1, e ¼0.6).
0.2
0.3
ε
0.4
0.5
0.6
Fig. 7. Relationship between attitude angle and eccentricity ratio under the influence of various power law indices and surface roughness (n ¼1, am ¼0.5).
95 Smooth, Λ = 0
90 85
Longitudinal, Λ = 0.3
80
Transverse, Λ = 0.3
75 70 65
ε = 0.3
60 fR/C
smooth bearing surfaces, whereas transverse roughness slightly reduce dimensionless load. Fig. 7 illustrates the relationship between attitude angle j and eccentricity ratio e under the influence of a magnetic field and various power law indices. Clearly, the attitude angle is markedly reduced with an increase in eccentricity ratio. With a fixed power law index, the influence of roughness pattern on attitude angle grows with an increase in eccentricity ratio. Taking shear thickening fluid (n¼1.5) as an example, longitudinal roughness can reduce the attitude angle by approximately 431 when eccentricity increases from 0.1 to 0.6. Under the same roughness pattern, lower power law indices lead to higher attitude angles. For this reason, shear thinning fluids (n¼0.7) possess higher attitude angles. Fig. 8 shows the relationship between the modified friction coefficient and the magnetic field coefficient under various eccentricity ratios. Taking e ¼0.3 as an example, the modified friction coefficient shows a distinctively decreasing trend as the magnetic field coefficient increases. However, in the case of a high eccentricity ratio (e ¼0.6), the change is slight. Longitudinal roughness reduces the modified friction coefficient, whereas transverse roughness slightly increase the modified friction coefficient. Fig. 9 exhibits the relationship between the modified friction coefficient and roughness parameters under the influence of a fixed magnetic field and various power law indices. Using n¼1.5 as an
55 50 45 40 35 30
ε = 0.6
25 20 15 0
0.1
0.2
αm
0.3
0.4
0.5
Fig. 8. Relationship between modified friction coefficient and magnetic field coefficient under various eccentricity ratios and surface roughness (n¼ 1).
fR/C
T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
175
increasingly significant as the eccentricity ratio rises. We hope that the simulation results in this study could provide a reference for the future design of bearings.
27 26.5 26 25.5 25 24.5 24 23.5 23 22.5 22 21.5 21 20.5 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 15 14.5
n = 0.7 References
n = 1.0 n = 1.5
Longitudinal Transverse
0
0.1
0.2 Λ
0.3
0.4
Fig. 9. Relationship between modified friction coefficient and roughness parameter under the influence of various power law indices and surface roughness (e ¼ 0.6, am ¼ 0.5).
example, the modified friction coefficient in a longitudinal roughness decreases as the roughness parameter increases; however, the modified friction coefficient in a transverse roughness increases. Furthermore, observations of the transverse roughness show that when L is between 0.35 and 0.37, a unique region of the exchange mechanism appears in the influence of power law index on the modified friction coefficient.
4. Conclusion This study investigated the influence of ferrofluids on the lubrication performance of short journal bearings under the combined effects of stochastic surface roughness and a magnetic field generated by a concentric finite wire. Film pressure distribution on the short journal bearing was calculated to predict vital parameters including loading capacity, attitude angle, and modified friction coefficient. The results were compared with those of relevant studies to verify the reliability of the theoretical structure and the numerical analysis. The combined influence of surface roughness and a magnetic field produced by the concentric finite wire on the operational performance of bearings is significant and cannot be overlooked. Compared to the lubrication performance of bearings with smooth surfaces, inducing a magnetic field and introducing longitudinal roughness can increase Pmax, and enhance the loading capacity of the bearing. In addition, the attitude angle and modified friction coefficient can be reduced. Transverse roughness, on the other hand, has the opposite effect on the operational performance of bearings. The aforementioned trends become
[1] Raj K, Moskowitz B, Casciari R. Advances in ferrofluid technology. Journal of Magnetism and Magnetic Materials 1995;149:174–80. [2] Scott W, Suntiwattana P. Effects of oil additives on the performance of a wet friction clutch material. Wear 1995;181–183:850–5. [3] Osman TA, Dorid M, Safar ZS, Mokhtar MOA. Experimental assessment of hydrostatic thrust bearing performance. Tribology International 1996;29(3):233–9. [4] Hong RY, Ren ZQ, Han YP, Li HZ, Zheng Y, Ding J. Rheological properties of water-based Fe3O4 ferrofluid. Chemical Engineering Science 2007;62:5912–24. [5] Oladeinde MH, Edokpia RO. Performance modeling of infinitely wide exponentially shaped slider bearing lubricated with couple stress fluids. The Journal of Engineering & Applied Sciences 2010;5:26–31. [6] Lin JR, Liang LJ, Chu LM. Effects of non-Newtonian micro-polar fluids on the squeeze-film characteristics between a sphere and a plate surface. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 2010;224:825–32. [7] Lin JR. Derivation of ferrofluid lubrication equation of cylindrical squeeze films with convective fluid inertia forces and application to circular disks. Tribology International 2012;49:110–5. [8] Mitamura Y, Arioka S, Sakota D. Application of a magnetic fluid seal to rotary blood pump. Journal of Physics: Condensed Matter 2008;20:2041–5. [9] Sihuan Y, Huang W, Wang X. Orientation effects of micro-grooves on sliding surface. Tribology International 2011;44:1047–54. [10] Katiyar A, Singh AN, Shukla P, Nandi T. Rheological of magnetic nanofluids containing spherical nanoparticles of Fe–Ni. Powder Technology 2012;224: 86–9. [11] Osman TA, Nada GS, Safar ZS. Effect of using current-carrying-wire models in the design of hydrodynamic journal bearings lubricates with ferrofluid. Tribology Letters 2004;11:61–70. [12] Osman TA, Nada GS, Safar ZS. Static and dynamic characteristics of magnetized journal bearings lubricated with ferrofluid. Tribology International 2001;34:369–80. [13] Osman TA, Nada GS, Safar ZS. Different Magnetic models in the design of hydrodynamic journal bearings lubricated with non-Newtonian ferrofluid. Tribology Letters 2003;14:211–23. [14] Osman TA. Effect of lubricant non-Newtonian behavior and elastic deformation on the dynamic performance of finite journal plastic bearings. Tribology Letters 2001;17:31–40. [15] Nada GS, Osman TA. Static performance of finite hydrodynamic journal bearings lubricated by magnetic fluids with couple stresses. Tribology Letters 2007;27:261–8. [16] Huang W, Shen C, Wang X. Study on static supporting capacity and tribological performance of ferrofluid. Tribology Transactions 2009;52: 717–23. [17] Zhu D. Effect of surface roughness on mixed EHL lubrication characteristics. Tribology Transactions 2003;46:44–8. [18] Christensen H. Stochastic models for hydrodynamic lubrication of rough surface. Journal and Proceedins of the Institute Mechanical Engineering 1969–1970;55:1013–25. [19] Christensen H, Tonder K. The hydrodynamic lubrication of rough bearing surface of finite width. Journal of Lubrication Technology 1971;93:324–30. [20] Hsu CH, Lin JR, Chiang HL. Combined effects of couple stresses and surface roughness on the lubrication of short journal bearings. Industrial Lubrication & Tribology 2003;55:233–43. [21] Chiang HL, Hsu CH, Lin JR. Lubrication performance of long journal bearings considering effects of couple stresses and surface roughness. Journal of the Chinese Institute of Chemical Engineers 2004;27(2):287–92. [22] Chiang HL, Hsu CH, Lin JR. Lubrication performance of finite journal bearings considering effects of couple stresses and surface roughness. Tribology International 2004;37:297–307. [23] Chiang HL, Lin JR, Hsu CH, Chang YP. Linear stability analysis of a rough short journal bearing lubricated with non-Newtonian fluids. Tribology Letters 2004;17:867–77. [24] Stokes VK. Couple stress in fluids. Physics of Fluids 1966;9:1709–15. [25] Hsu CH, Lai C, Lu RF, Lin JR. Combined effects of surface roughness and rotating inertia the squeeze characteristics of parallel circular disk. Journal of Mechanical Science and Technology 2009;17:60–6.
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Lubrication performance of short journal bearings considering the effects of surface roughness and magnetic field Tze-Chi Hsu a,n, Jing-Hong Chen a, Hsin-Lu Chiang b, Tsu-Liang Chou b a b
Department of Mechanical Engineering, Yuan-Ze Unversity, Chung-Li 320, Taiwan, ROC Department of Mechanical Engineering, Nanya Institute of Technology, PO Box 267, Chung-Li 320, Taiwan, ROC
a r t i c l e i n f o
abstract
Article history: Received 22 May 2012 Received in revised form 16 November 2012 Accepted 11 December 2012 Available online 11 January 2013
The primary objective of this study was to investigate the performance of ferrofluids under the combined influence of surface roughness and a magnetic field generated by a concentric finite wire. The distribution of pressure was obtained and used to calculate the characteristics of the bearing. The results of this study indicate that the combined influence affect the distribution of film pressure, which enhances loading capacity and reduces the modified friction coefficient. The effect of longitudinal roughness was particularly significant and we expect these findings could further enhance the design of short journal bearing within a magnetic field. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Short journal bearing Ferrofluid Christensen stochastic roughness theory Concentric finite wire magnetic field
1. Introduction Ferrofluids are a non-Newtonian fluid with the strong magnetism of solid magnetic materials and the fluidity of liquids. A number of studies [1–6] have shown that ferrofluids have exceptional lubricating characteristics capable of improving the load capacity of bearings, reducing wear, and lengthening the operational life. Ferrofluids have been studied in various practical applications, including the finishing of optical instruments, fluid clutches, sealants, and in automotive and civil damping [7–10]. Osman et al. [11–15] investigated the influence of ferrofluid lubricants on the operational characteristics of journal bearings. In addition to deriving generalized ferrofluid Reynolds equations, they investigated the effects of ferrofluids using specifically designed magnetic field models. Their results have indicated that increasing the power law index of ferrofluids and the intensity of magnetic fields can enhance the loading capacity and attitude angle of the bearing and decrease the modified friction coefficient of the bearing in the case of high eccentricity. However, the model of the applied magnetic field must be selected with care to prevent the occurrence of side leakage due to excessive film pressure. Huang et al. [16] verified through experimentation that Fe3O4 ferrofluid could be used to improve the static loading
n Correspondence to: Department of Mechanical Engineering, Yuan-Ze Unversity 135 Yuan-Tung Road, Chung-Li 320, Taiwan, ROC. Tel.: þ 886 3 4638800x2458; fax: þ 886 3 4558031. E-mail addresses: [email protected], [email protected] (T.-C. Hsu).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2012.12.016
capacity and friction characteristics of bearings under the influence of an external magnetic field. These studies all hypothesized that the surfaces of the bearings are completely smooth. Zhu [17] showed that manufacturing processes often produce microscopic surface features displaying transverse, longitudinal, and isotropic surface roughness. Christensen and Tonder [18,19] developed stochastic Reynolds equations to describe average pressure, addressing transverse and longitudinal roughness, and determined the influence of roughness on the performance of bearings. Chiang et al. [21–23] derived a generalized stochastic Reynolds equation based on Christensen’s stochastic model and the Stokes microcontinuum theory [24]. Through numerical simulation, the performances of lubrication in bearings under the combined influence of couple stress and surface roughness was determined to characterize the dynamic squeeze film. Hsu et al. [25] applied Christensen’s stochastic roughness theory between two parallel circular disks, their results of which demonstrate that surface roughness could improve squeeze film behavior. Magnetic particles are often trapped in the valleys associated with surface roughness; however, previous studies have tended to focus on the impact of surface roughness, without considering the influence of magnetic fields. The objective of the present study was to investigate the combined effects of surface roughness as well as magnetic fields. This study addressed short journal bearings with a length-todiameter ratio of r0.25. A magnetic field was created using a concentric metal wire with finite length to investigate the operational characteristics of a short journal bearing lubricated with non-Newtonian power law ferrofluid under the influences of
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Mg n Pm p, P R Xm x, y, z Z
Namenclature C C e fMX fMZ hm hm0 Hm h hm, H L m0
One half of the total range of the random film thickness variable Radial clearance Eccentricity, e ¼ eC Magnetic forces in x-direction (circumferential direction) Magnetic forces in z-direction (axial direction) Magnetic field intensity Characteristic value of magnetic field intensity Dimensionless magnetic field intensity, HM ¼ hM =hM0 Thickness of lubricant film Nominal smooth part of the film thickness, H¼hm/ C ¼1þ ecosy Length of the bearing Viscosity constant
am e d y l L
m0 o
Magnetization of ferrofluid Power-law index Induced magnetic pressure n þ 1 Mean film pressure, P ¼ ðp==m0 on ÞðÞ C=R Radius of the journal Susceptibility of ferrofluid Rectangular coordinates Dimensionless coordinate in the z-direction, Z¼ z/L Magnetic field coefficient, am ¼ ðh2M0 m0 X m =m0 on ÞðC=RÞn þ 1 Eccentricity ratio, e ¼ e=C Random part of film geometry Circumfer ential coordinate, x ¼Ry Length-to-diameter, l ¼ L=2R Roughness parameter, L ¼c/C Permeability of free space of air, m0 ¼4p 10 7 AT/m Angular speed
n
Fig. 1 depicts the configuration of a short bearing with a journal radius R rotating at angular speed o, with lubrication of the interior bearings provided by ferrofluid. A finite metal wire is located in the center of the bearing, through which electric current I induces a magnetic field. Based on the Navier–Stokes equation, the magnetic force was treated as a external body force and the modified Reynolds equation was then obtained as [14] nþ2
f MX
þ
@ nþ2 h f MZ @z
ð1Þ
where n is the power-low index, p is local film pressure, and m0 is the viscosity constant. The fMX and fMZ are the components of magnetic force in the circumferential and axial directions, respectively, written as follows:
2. Theoretical analysis
@ h @x n
!
nþ2
@h @ h ¼ 6m0 ðRoÞ þ @x @x n
bearing surface roughness. It is hoped that our establishment of theoretical model could be used as a reference for the future design of bearings.
f MX ¼ m0 X M hM
@hM @x
ð2Þ
f MZ ¼ m0 X M hM
@hM @z
ð3Þ
here m0 is permeability of free space of air, m0 ¼4p 10 7 AT/m. XM is the susceptibility of ferrofluid. In the concentric field model with a finite wire the intensity of the magnetic field hM provides a magnetic field represented by [14] I L=2 þ z L=2z þ sintan1 ð4Þ sintan1 hM ðzÞ ¼ 4pR R R
! @p @ n þ 2 @p þ h @x @z @z
where I is the strength of the current passing through the wire.
cutaway view y θ
y
x
L
Concentric finite wire
z e
h
R
U
I I
x
h hm 1
A. Transverse Roughness
= 1 + 2 2 = 12 + 22
2
Fig. 1. Configuration of a journal bearing.
B. Longitudinal Roughness
T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
According to Christensen’s theorem using expected values of Eq. (1), the stochastic modified Reynolds equation for the journal bearing with rough surface can be written as: " ! #
nþ2 @ h @p @ n þ 2 @p E E h þ @x @x @z @z n " # ! nþ2 @EðhÞ @ h @ n þ 2 þ E E h f MZ ð5Þ f MX þ ¼ 6m0 ðRoÞn @x @x @z n where the expectancy operator E( ) is defined by Z 1 EðUÞ ¼ ðUÞf ðdÞdd
The non-dimensional modified Reynolds equation for longitudinal roughness can then be expressed as
@ @P n @ @P @H G2 ðH, L,nÞ ¼ 6n þ 2 G1 ðH, L,nÞ @y @y @Z @Z @y 4 l
@ @HM nam @ @HM G ð15Þ þ am G1 ðH, L,nÞHM ð H, L ,n ÞH þ M 2 2 @y @y @Z 4l @Z where hm C
ð16Þ
Z ¼ z=L
ð17Þ
L 2R
ð18Þ
y ¼ x=R
ð19Þ
H¼ ð6Þ
171
1
and f(d) is the probability density distribution for the stochastic variables. As most engineering rough surfaces are Gaussian in nature, one can choose for simplicity, instead of using a Gaussian distribution function for integration, a polynomial form as following: ( 2 35 ðc2 d Þ3 if c r d r c f ðdÞ ¼ 32c7 ð7Þ 0 elsewhere where c is the half total range of random film thickness variable and, the function terminates at c¼ 73s; where s is the standard deviation. In this study, local film geometry of the lubricant is treated as a stationary, ergodic, stochastic process with zero mean, written as h ¼ hm ðx,zÞ þ dðx,z, xÞ
ð8Þ
where hm represents the nominal smooth part of the film geometry depending upon the coordinates x and z, while d(x,z,x) is the part due to the surface asperities measured from the nominal level and is regarded as a randomly varying quantity of zero mean. Using the circumferential coordinate, the film thickness hm can also be expressed as hm ¼ C ð1 þ e cosyÞ
ð9Þ
where C is the radial clearance, and e is eccentricity ratio, e ¼ e=C. The surface feature of one-dimensional longitudinal roughness is assumed to have the form of long narrow ridges and valleys running in the direction of rotation. Thus, the thickness of the lubricating film listed in Eq. (8) can be expressed as a function of the form h ¼ hm ðx,zÞ þ dðz, xÞ
ð10Þ
whereupon Eq. (5) for longitudinal roughness can be reduced to 2 3 " ! # nþ2 @ h @p @ 4 1 @p 5 E þ @x @x @z E 1=hn þ 2 @z n 0 1 " # ! nþ2 @E ð h Þ @ h @ 1 f A þ E ¼ 6m0 ðRoÞn f MX þ @ @x @x @z E 1=hn þ 2 MZ n
l¼
G1 ðH, L,nÞ ¼ Hn þ 2 þ
G2 ðH, L,nÞ ¼
1 H
nþ2
1 n 2 2 H L n þ 3n þ2 18
þ
L2 ½n2 þ 5n þ 6 18 Hn þ 4
ð20Þ
ð21Þ
and the intensity of the dimensionless magnetic field of Eq. (4) can be written as
HM ðZÞ ¼ sin tan1 l þ2lZ þsin tan1 l2lZ ð22Þ where HM ¼ hM =hM0
ð23Þ
and hM0 ¼ I=4pR
ð24Þ
One-dimensional transverse roughness is assumed to have the form of long narrow ridges and valleys running in the direction perpendicular to rotation. Therefore, the lubricant film thickness can be expressed as a function of the form: h ¼ hm ðx,zÞ þ dðx, xÞ Eq. (15) for transverse roughness can be derived as
@ @P n @ @P G1 ðH, L,nÞ G2 ðH, L,nÞ þ 2 @y @y @Z 4l @Z
@ @ @HM ¼ 6n G:3 ðH, L,nÞ þ am G2 ðH, L,nÞHM @y @y @y
nam @ @HM G1 ðH, L,nÞHM þ 2 @Z 4l @Z
ð25Þ
ð26Þ
where G3 ðH, L,nÞ ¼ H
18H2 þ L2 ½n2 þ5n þ6 18H2 þ L2 ½n2 þ 7n þ 12
ð27Þ
where p is mean film pressure. To simplify the analysis, the magnetic field coefficient can be defines as n þ 1 2 h m X C am ¼ M0 0 n M ð12Þ R m0 o
Because l{ 1 for the approximation of a short journal bearing, the circumferential variation in pressure can be neglected in favor of axial variations. Therefore, Eqs. (15) and (26) are reduced to
@ @P 2 @H G2 ðH, L,nÞ ¼ 24l @Z @Z @y
@ @HM ðZÞ G2 ðH, L,nÞHM ðZ Þ Longitudinal ð28Þ þ am @Z @Z
and dimensionless mean film pressure P and surface roughness parameter L, written as n þ 1 p C ð13Þ P¼ m0 on R
@ @P @ 2 G1 ðH, L,nÞ ¼ 24l G3 ðH, L,nÞ @Z @Z @y
@ @HM ðZÞ G1 ðH, L,nÞHM ðZ Þ Transvers þ am @Z @Z
ð11Þ
L¼
c C
ð29Þ
The boundary conditions for the film pressure are as follows: ð14Þ
P¼0
at
Z ¼ 71=2
ð30Þ
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T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
dP ¼0 dZ
at
Z¼0
0.55
ð31Þ
0.5
applying the boundary conditions (30) and (31), the non-dimensional mean film pressure can be obtained by integrating Eqs. (28) and (29) as 2
P ¼ 12 l e sin y
1 1 am þ Z2 K ðZ Þ 2 G2 ðH, L,nÞ 4
0.45 0.4
Longtudinal
0.35
ð32Þ 2
P ¼ 12l
@ 1 1 am G3 ðH, L,nÞ Z2 K ðZ Þ þ 2 @y G1 ðH, L,nÞ 4
0.3 Transverse
P 0.25
ð33Þ
0.2
where ð34Þ
0.15
By integrating the non-dimensional mean film pressure acting on the journal bearing, the dimensionless load components parallel and perpendicular to the line of centre W0 and Wp/2 can be acquired by Z p 1 2 sin y W 0 ¼ 2l e 0 G2 ðH, L,nÞ Z pZ 1 2 cos y dyam K 1 ðZÞcos ydZdy Longitudinal ð35Þ
0.1
KðZÞ ¼ ðHM ðZÞÞ2 ðHM ðZÞ9Z ¼ 0:5 Þ2
0
2
W 0 ¼ 2l
Z p @
0
G3 ðH, L,nÞ
@y Z pZ
0
1 2
cos y dyam
0
2
Wp=2 ¼ 2l e þ am
Z p 0
Z pZ 0
1 2
1 G1 ðH, L,nÞ
K 1 ðZÞcos ydZdy
Transverse
ð36Þ
1 sin2 ydy G2 ðH, L,nÞ Longitudinal
ð37Þ
0
2
Wp=2 ¼ 2l
Z p @ 0
sin ydy þ am
G3 ðH, L,nÞ
@y Z pZ 0
1 2
1 G1 ðH, L,nÞ
K 1 ðZÞsin ydZdy
Transverse
ð38Þ
0
Thereafter, the dimensionless load capacity W and attitude angle j can be evaluated from above, as follows: W ¼ ½W 0 2 þ W p=2 2 1=2
ð39Þ
j ¼ tan1 ½W p=2 =W 0
ð40Þ
The shear stress on the moving surface in the direction of rotation can also be calculated, such that the modified friction coefficient f R/C is determined by
2 f R=C ¼
f R=C ¼
R 12 R 2p 0
0
1 Hn
0
60
90 θ
120
150
180
The objective of this study was to investigate the influence of ferrofluids on the lubrication performance of short journal bearing systems under the combined effects of surface roughness and a magnetic field model generated by a finite concentric wire. To verify the accuracy of the derived theory and the reliability of the numerical analysis, we compare our results with those of Hsu et al. [20] and Osman et al. [12]. Hsu et al. [20] focused on the influence of surface roughness in journal bearing systems. Under a magnetic force of 0, our numerical analysis of the smooth case is similar to that obtained by Chiang. Furthermore, the increase in the magnetic field contributed to an increase in lubricant pressure, as shown in Fig. 2. On the other hand, Osman et al. [12] focused on the influence of magnetism on bearing characteristics without considering the effects of surface roughness. Fig. 3 presents a comparison of the current simulations and those of Osman (adopting the same assumptions), in which similar results were obtained. The impact of surface roughness was also observed in the current model, in which longitudinal roughness facilitated an increase in Pmax, whereas transverse roughness had the opposite effect. Although the viscosity of ferrofluids changes only slightly in the Schliomis model, a number of studies [11–15] have shown that viscosity can be affected by differences in the weight ratio of
W n h ion R 1 R 2p 2 2 02 0 H1n 1þ g 12l Z 2 14 @@y @@y G3 ðH, L,nÞ G1 ðH,1L,nÞ dydZ
ð1 þ ecos yÞn þ 1 2n
30
Fig. 2. Comparison between current simulations with Hsu et al. [20] (Z¼ 0, n¼ 1, e ¼ 0.6, l ¼ 0.25, L ¼ 0).
h in 2 1 þ g ð12Þ l e Z 2 14 @@y sin y G2 ðH,1L,nÞ dy dZ
W
where
g¼
0
3. Results and discussion
0
K 1 ðZÞsinydZdy
0.05
ð43Þ
Longitudinal
Transverse
ð41Þ
ð42Þ
ferromagnetic particles within the fluid. Therefore, the impact of various power law indices on dimensionless pressure distribution under the influence of surface roughness is shown in Fig. 4. Clearly, with an increase in the power law index, using longitudinal roughness as an example, shear thickening fluid (n¼1.5) has the highest
T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
0.65
2.1 Osman, 2001
1.95
0.6
Smooth, Λ = 0
1.8
Longitudinal, Λ = 0.3
0.55
Transverse, Λ = 0.3
1.65
0.5
1.5
0.45
1.35
0.4
1.2
0.35
1.05
P
P
173
0.3
0.9
0.25
0.75
αm = 0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 αm = 0.5,Smooth,Λ = 0 Longitudinal,Λ = 0.3 Transverse, Λ = 0.3
0.2
0.6 0.45
0.15
0.3
0.1
0.15
0.05
0
0
0
30
60
90
θ
120
150
180
210
Fig. 3. Comparison between current simulations with Osman et al. [12] (Z¼ 0, n¼ 1, e ¼0.5, l ¼ 1, am ¼ 0).
1.1
n = 0.7, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.5, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3
1 0.9 0.8 0.7
P
0.6 0.5 0.4 0.3 0.2 0.1 0 0
30
60
90 θ
120
150
180
Fig. 4. Relationship between dimensionless pressure P and y under the influence of various power law indices and surface roughness (Z ¼0, e ¼ 0.6, l ¼0.25, am ¼ 0.5).
Pmax, followed by Newtonian fluid (n¼1.0) and shear thinning fluid (n¼0.7). Fig. 5 shows the relationship between dimensionless pressure P and Z at y ¼1501 under different magnetic field coefficient. It is clear that increasing the coefficient of the magnetic field derives higher dimensionless pressure in which, Pmax increases considerably. In addition, the same trend is observed that the surface roughness has a significant influence on the distribution of dimensionless pressure P. With am ¼0.5 as an example, longitudinal roughness increased Pmax compared to a smooth bearing surface whereas transverse roughness slightly decreased Pmax.
-0.5
-0.4
-0.3
-0.2
-0.1
0 Z
0.1
0.2
0.3
0.4
0.5
Fig. 5. Relationship between dimensionless pressure P and y under various magnetic field effects and surface roughness (y ¼ 1501, n ¼1, e ¼0.6, l ¼ 0.25).
Table 1 illustrates the combined effects of surface roughness and a magnetic field according to variations in Pmax. When the magnetic field coefficient equaled zero, the maximum dimensionless pressure increased to 17.8%, in conjunction with an increase in longitudinal roughness from L ¼0 to L ¼0.3. In contrast, an increase in transverse roughness slightly decreased the maximum dimensionless pressure, compared to the smooth case. The current study assumed that the short journal bearings had a length-to-diameter ratio equal to 0.25; therefore, circumferential variations in pressure were neglected in favor of those in the axial direction. As a result, the longitudinal grooves generated stronger hydrodynamic lift, resulting in higher maximum dimensionless pressure; however, as the length-to-diameter ratio increased, the dominance of surface roughness patterns on dimensionless pressure changed from longitudinal to transverse. Research is currently underway to investigate this phenomenon with regard to long journal bearings. The influence of a magnetic field on maximum dimensionless pressure is presented in Table 1. Because the surface was assumed to be smooth, the maximum dimensionless pressure increased to 6.7% when the magnetic field coefficient was increased from 0 to 0.5. Finally, combining the influence of surface roughness and magnetic field resulted in a maximum dimensionless pressure of 28.4%, in the case of L ¼0.3 and am ¼0.5. It is clear that with an increase in the magnetic field, the influence of surface roughness is far more pronounced than that of a magnetic field and the coupled effects become more significant. To ensure strong hydrodynamic lift within the journal bearing and prevent side leakage caused by excessive hydrodynamic pressure, an appropriate balance must be attained between these two factors. Fig. 6 exhibits the relationship between dimensionless load W and eccentricity ratio e under various magnetic field coefficients. Clearly, dimensionless load is directly proportional to eccentricity ratio. The increase of dimensionless load due to the increase of magnetic field coefficient is more apparent in the low eccentricity ratio. Moreover, the surface roughness pattern also has an effect on dimensionless load. Taking am ¼0.5 as an example, longitudinal roughness can increase dimensionless load, in comparison with
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T.-C. Hsu et al. / Tribology International 61 (2013) 169–175
Table 1 Pmax as a function of surface roughness at various am (Z ¼0, n ¼1, e ¼ 0.6, l ¼ 0.25). Surface roughness
Magnetic field coefficient am ¼0
Longitudinal L ¼ 0.3 L ¼ 0.1 Smooth L¼0 Transverse L ¼ 0.3 L ¼ 0.1
Pmax 0.597 0.557 0.507 0.492 0.475
Pmax, L P max, am ¼ 0, L ¼ 0 P max, am ¼ 0, L ¼ 0
Pmax, L P max, am ¼ 0, L ¼ 0 P max, am ¼ 0, L ¼ 0
100%(%) Magnetic field coefficient am ¼0.2 Pmax 0.615 0.571 0.518 0.504 0.486
17.8 9.9 0 3.0 6.3
100%(%) Magnetic field coefficient am ¼ 0.5 Pmax 0.651 0.601 0.541 0.528 0.51
21.3 12.6 2.2 0.6 4.1
Pmax, L P max, am ¼ 0, L P max, am ¼ 0, L ¼ 0
¼ 0
100%(%)
28.4 18.5 6.7 4.1 0.6
90
0.45 Smooth, 0.4
Longitudinal, Λ = 0.3
85
Transverse, Λ = 0.3
80
0.35
75
0.3 70 ϕ
W
0.25 65
0.2
n = 0.7,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.0,Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3 n = 1.5, Smooth, Λ = 0 Longitudinal, Λ = 0.3 Transverse, Λ = 0.3
60
0.15 55
αm = 0.5
0.1
50
αm = 0 0.05
45
0
40
0.1
0.2
0.3
0.4
0.5
0.6
0.1
ε Fig. 6. Relationship between dimensionless load and eccentricity ratio under various magnetic field effects and surface roughness (n¼ 1, e ¼0.6).
0.2
0.3
ε
0.4
0.5
0.6
Fig. 7. Relationship between attitude angle and eccentricity ratio under the influence of various power law indices and surface roughness (n ¼1, am ¼0.5).
95 Smooth, Λ = 0
90 85
Longitudinal, Λ = 0.3
80
Transverse, Λ = 0.3
75 70 65
ε = 0.3
60 fR/C
smooth bearing surfaces, whereas transverse roughness slightly reduce dimensionless load. Fig. 7 illustrates the relationship between attitude angle j and eccentricity ratio e under the influence of a magnetic field and various power law indices. Clearly, the attitude angle is markedly reduced with an increase in eccentricity ratio. With a fixed power law index, the influence of roughness pattern on attitude angle grows with an increase in eccentricity ratio. Taking shear thickening fluid (n¼1.5) as an example, longitudinal roughness can reduce the attitude angle by approximately 431 when eccentricity increases from 0.1 to 0.6. Under the same roughness pattern, lower power law indices lead to higher attitude angles. For this reason, shear thinning fluids (n¼0.7) possess higher attitude angles. Fig. 8 shows the relationship between the modified friction coefficient and the magnetic field coefficient under various eccentricity ratios. Taking e ¼0.3 as an example, the modified friction coefficient shows a distinctively decreasing trend as the magnetic field coefficient increases. However, in the case of a high eccentricity ratio (e ¼0.6), the change is slight. Longitudinal roughness reduces the modified friction coefficient, whereas transverse roughness slightly increase the modified friction coefficient. Fig. 9 exhibits the relationship between the modified friction coefficient and roughness parameters under the influence of a fixed magnetic field and various power law indices. Using n¼1.5 as an
55 50 45 40 35 30
ε = 0.6
25 20 15 0
0.1
0.2
αm
0.3
0.4
0.5
Fig. 8. Relationship between modified friction coefficient and magnetic field coefficient under various eccentricity ratios and surface roughness (n¼ 1).
fR/C
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175
increasingly significant as the eccentricity ratio rises. We hope that the simulation results in this study could provide a reference for the future design of bearings.
27 26.5 26 25.5 25 24.5 24 23.5 23 22.5 22 21.5 21 20.5 20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 15 14.5
n = 0.7 References
n = 1.0 n = 1.5
Longitudinal Transverse
0
0.1
0.2 Λ
0.3
0.4
Fig. 9. Relationship between modified friction coefficient and roughness parameter under the influence of various power law indices and surface roughness (e ¼ 0.6, am ¼ 0.5).
example, the modified friction coefficient in a longitudinal roughness decreases as the roughness parameter increases; however, the modified friction coefficient in a transverse roughness increases. Furthermore, observations of the transverse roughness show that when L is between 0.35 and 0.37, a unique region of the exchange mechanism appears in the influence of power law index on the modified friction coefficient.
4. Conclusion This study investigated the influence of ferrofluids on the lubrication performance of short journal bearings under the combined effects of stochastic surface roughness and a magnetic field generated by a concentric finite wire. Film pressure distribution on the short journal bearing was calculated to predict vital parameters including loading capacity, attitude angle, and modified friction coefficient. The results were compared with those of relevant studies to verify the reliability of the theoretical structure and the numerical analysis. The combined influence of surface roughness and a magnetic field produced by the concentric finite wire on the operational performance of bearings is significant and cannot be overlooked. Compared to the lubrication performance of bearings with smooth surfaces, inducing a magnetic field and introducing longitudinal roughness can increase Pmax, and enhance the loading capacity of the bearing. In addition, the attitude angle and modified friction coefficient can be reduced. Transverse roughness, on the other hand, has the opposite effect on the operational performance of bearings. The aforementioned trends become
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