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Effects of surface roughness and non-Newtonian micropolar fluids on dynamic characteristics of wide plane slider bearings
Tribology International 66 (2013) 150–156
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Short Communication
Effects of surface roughness and non-Newtonian micropolar fluids on dynamic characteristics of wide plane slider bearings Jaw-Ren Lin a,n, Tzu-Chen Hung b, Tsu-Liang Chou a, Long-Jin Liang a a b
Department of Mechanical Engineering, Taoyuan Innovation Institute of Technology, Jhongli, Taiwan, ROC Department of Mechanical Engineering, National Taipei University of Technology, Taipei, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 16 February 2013 Received in revised form 5 May 2013 Accepted 7 May 2013 Available online 14 May 2013
Based on the Eringen's microcontinuum theory and the Christensen's stochastic theory, the combined effects of non-Newtonian rheology and surface roughness on the dynamic characteristics of slider bearings have been investigated. According to the results, the effects of transverse roughness provide an increase in the load capacity and dynamic coefficients as compared to the smooth bearing lubricated with micropolar fluids, whereas the influences of longitudinal roughness yield a reversed trend. The quantitative effects of rough surfaces and non-Newtonian fluids on bearing performances are more pronounced for higher values of the roughness parameter, the coupling parameter and the interacting parameter. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Surface roughness Non-Newtonian micropolar fluids Plane slider bearing Dynamic coefficients
1. Introduction Hydrodynamic slider bearings are commonly designed to sustain a transverse load in rotating machine systems. The performances of slider bearings have been investigated under various operating conditions, for example, the consideration of turbulent flows by Taylor and Dowson [1] and Capitao [2]; the inclusion of dynamic squeezing action by Lin et al. [3]; the inertia force effects by Launder and Leschziner [4] and Elrod et al. [5]; the thermal and inertia influences by Talmage and Carpino [6]; and the thermal turbulent effects by Shyu et al. [7]. All of these contributions focus on the use of a Newtonian lubricant. According to the experimental study of Scott and Suntiwattana [8], the use of additives provides a significant improvement in reducing the wear of wet clutch plates. Therefore, various types of additives are often added in lubricants to improve their performances. Since the conventional Newtonian theory is not adequate to describe these kinds of fluids containing substructures, a microcontinuum theory of micropolar fluids has been developed by Eringen [9]. The micropolar fluids can define spin inertia and microrotational effects generated by the local structures. According to the study of Eringen [9], micropolar fluids are important for fluids containing small particles, such as liquid crystals, animal bloods, polymetric suspensions and fluids blended additives. Therefore, the
n Corresponding author. No. 414, Sec. 3, Jhongshan E. Rd., Jhongli, 320, Taiwan, ROC. Tel.: +886 3 4361070x6212; fax: +886 3 4384670. E-mail address: [email protected] (J.-R. Lin).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2013.05.004
theory of micropolar fluids has been applied to investigate the performance characteristics of various bearing systems, such as the capillary compensated hydrostatic bearings by Verma et al. [10], the orifice compensated hydrostatic/hybrid bearings by Nicodemus and Sharma [11], the orifice compensated nonrecessed hole-entry hybrid bearings by Ram and Sharma [12], the journal bearings by Gorla and Nashery [13] and Das et al. [14]; the squeeze film bearings by Tandon and Jaggi [15], Sinha and Singh [16] and Tandon and Chaurasia [17]; and the slider bearings by Ramanaiah and Dubey [18], Isa and Zaheeruddin [19], Agrawal and Bhat [20], Bayada et al. [21], Naduvinamani and Marali [22] and Lin et al. [23]. All of these contributions investigate the lubrication performances assuming that the bearing surfaces are perfectly smooth. However, bearing surfaces may be roughened owing to the manufacturing process. Therefore, the influences of surface roughness on bearing performances should be included. Applying the concept of a stochastic process, a modified Reynolds equation for rough bearing surfaces is derived by Christensen [24]. The longitudinal and transverse surface roughness effects on the operating characteristics of plane slider bearings are analyzed. It is reported that surface roughness may considerably influence the bearing characteristics. Based on the stochastic theory of Christensen [24], many contributions to the study of rough bearings have been presented, such as the investigations in the hydrostatic bearings by Lin [25]; the journal bearings by Turaga et al. [26] and Guha [27]; and the slider bearings by Christensen and Tonder [28], Tonder [29] and Andharia et al. [30]. According to their results, the consideration of surface roughness results in significant influences on the bearing characteristics. Recently, the steady performance of
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
Nomenclature A,B d
Dd,Dnd E F,F* h,h* h1 hm0 n hm ; hm n
hs ; hs H,H* k l L N
length and width of the bearing shoulder height 3 dynamic damping coefficient, Dnd ¼ Dd hm0 =μA3 B (dimensionless) expectancy operator 2 n dynamic film force, F n ¼ Fhm0 =μUA2 B ¼ F n ðhm ; V n Þ n n n n n smooth part of film profile, h ðx ; t Þ ¼ h=hm0 ¼ h þ hm (dimensionless) inlet film thickness steady outlet film thickness n outlet film thickness, hm ðt n Þ ¼ hm ðtÞ=hm0 (dimensionless) n mean film shape, hs ¼ hs =hm0 ¼ ð1−xn Þδ local film thickness, H* ¼H/hm0 ¼h*+α* (dimensionless) vortex viscosity coefficient of micropolar fluids characteristic material length, l ¼(γ/4μ)1/2 interacting parameter, L¼l/hm0 (dimensionless) coupling parameter, N ¼ [k/(2μ+k)]1/2 (dimensionless)
rough inclined stepped composite non-porous and porous slider bearings with micropolar fluids have been analyzed by Naduvinamani and Siddangouda [31] and Siddangouda [32], respectively. Since few studies consider the combined effects of surface roughness and non-Newtonian micropolar fluids on the dynamic characteristics of slider bearings, a further investigated is motivated. The aim of the present study is mainly concerned with the combined effects of surface roughness and micropolar fluids on the dynamic characteristics of plane slider bearings. Based on the microcontinuum theory of Eringen [9] incorporating the stochastic theory of Christensen [24], a stochastic non-Newtonian dynamic Reynolds equation is derived in the present study. Analytical expressions of the steady load capacity and the dynamic coefficients are obtained. The steady performance and the dynamic characteristics of the rough bearing lubricated with micropolar fluids are presented and discussed through the variation of the roughness parameter, the coupling parameter and the interacting parameter as compared to the smooth bearing lubricated with a Newtonian fluid. Some numerical values of plane bearing performances are also presented in tables for engineering references.
P P p* Sd,Snd t,t* u U V* W,W* x, y, z x* α γ δ λ Λ μ
151
local film pressure mean film pressure, P ¼ EðPÞ 2 mean film pressure, pn ¼ Phm0 =μUA (dimensionless) 3 mean stiffness coefficient, Snd ¼ Sd hm0 =μUA2 B (dimensionless) time, t* ¼Ut/A (dimensionless) velocity components in the x-direction sliding velocity of the lower part n squeezing velocity, V n ¼ dhm =dt n (dimensionless) 2 mean steady load capacity, W n ¼ Whm0 =μUA2 B (dimensionless) Cartesian coordinates horizontal coordinate, xn ¼ x=A (dimensionless) random part resulting from surface asperities spin gradient viscosity coefficient of micropolar fluids shoulder parameter, δ ¼ d=hm0 (dimensionless) half total range of random film thickness variable roughness parameter, Λ ¼ λ=hm0 (dimensionless) classical viscosity coefficients of Newtonian fluids
slider bearings lubricated with a micropolar fluid governing the local film pressure P can be written as derived by Lin et al. [23]: ∂ ∂P ∂H gðH; N; lÞ −6μUH ¼ 12μ ð3Þ ∂x ∂x ∂t where 2
gðH; N; lÞ ¼ H 3 þ 12l H−6NlH 2 coth½NH=ð2lÞ
ð4Þ
l ¼ ½γ=ð4μÞ1=2 ;
ð5Þ
N ¼ ½k=ð2μ þ kÞ1=2
In this equation, μ is the classical viscosity coefficient, γ is the spin gradient viscosity coefficient and k is the vortex viscosity coefficient responsible for micropolar fluids. In addition, l denotes the characteristic material length, N represents the dimensionless coupling parameter. Application of the stochastic approach of rough surfaces of Christensen [24] and taking the expected values for both sides of Eq. (3), one can obtain the stochastic non-Newtonian dynamic Reynolds equation: ∂ ∂P ∂EðHÞ E gðH; N; lÞ −6μUH ¼ 12μ ð6Þ ∂x ∂x ∂t
2. Analysis Fig. 1 shows the physical geometry of a wide inclined plane slider bearing with rough surfaces. The lubricant is taken to be a non-Newtonian incompressible micropolar fluid of Eringen [9]. The local film thickness H can be considered to be composed of two separated parts: H ¼ hðx; tÞ þ αðx; y; ξÞ
ð1Þ
where h(x,t) describes the nominal smooth part of the film profile depending on the horizontal coordinate x and the time t: hðx; tÞ ¼ hs ðxÞ þ hm ðtÞ ¼ ðA−xÞ
d þ hm ðtÞ A
ð2Þ
On the other hand, α(x,y,ξ) represents the random part as a result of the surface asperities measured from the nominal mean level, and ξ denotes a random variable describing a definite roughness arrangement. Assume that body forces and body couples are neglected and the usual thin film lubrication theory is applicable, the non-Newtonian dynamic Reynolds equation for
Fig. 1. Physical geometry of a wide rough plane slider bearing lubricated with a non-Newtonian micropolar fluid.
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In this equation, E() denotes the expectancy operator defined by
Z
EðÞ ¼
∞ α ¼ −∞
ðÞf ðαÞdα
Longitudinal : pn ¼ 6
where λ describes the half total range of random film thickness variable. In the present study, it is mainly concerned with two types of roughness patterns: the longitudinal and transverse roughness structures. For one-dimensional longitudinal and transverse structures, the roughness patterns possess the form of narrow ridges and valleys running in the x and y-direction, respectively: Longitudinal : H ¼ hðx; tÞ þ αðy; ξÞ
Transverse : pn ¼ 6
ð10Þ
The stochastic non-Newtonian dynamic Reynolds equations are (
∂P E gðH; N; lÞ ∂x
(
1 ∂P E½g −1 ∂x
)
)
∂EðHÞ ∂EðHÞ þ 12μ ¼ 6μU ∂x ∂t
¼ 6μU
∂ E½H=gÞ ∂x E½1=g
ð11Þ
þ 12μ
∂EðHÞ ∂t
Z
n
G1 ðh ; N; L; ΛÞ ¼
ð13Þ n
h ¼ n
h α H n n n ¼ hs þ hm ; αn ¼ ; Hn ¼ ¼ h þ αn hm0 hm0 hm0 n
g ðH ; N; LÞ ¼
g (
n
f ðαn Þ ¼ hm0 f ¼
δ¼
¼H
3
hm0
2
n3
þ 12L H −6NLH coth NH n =ð2LÞ
n2 3
2
n
n2
35ðΛ −α Þ 32Λ7
if −Λ ≤αn ≤ þ Λ
0
elsewhere
d λ l ; Λ¼ ; L¼ hm0 hm0 hm0
ð14Þ ð15Þ
∂ ∂xn n
1 ∂pn E½g n−1 ∂xn
Λ
Z
Λ
¼6
∂ ∂xn
E½H n g n−1 þ 12V n n −1 E½g
1 g n ðH n ; N; LÞ
−Λ
ð16Þ
ð19Þ
where V n ¼ dhm =dt n denotes the squeezing velocity. Performing the integrations in the expectancy operator and applying the boundary conditions p*(x* ¼0) ¼0 and p*(x* ¼1) ¼0, one can obtain the dimensionless dynamic film pressure.
Z
xn
xn
G3 xn dxn
0
G3 dxn
0
ð21Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð22Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð23Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð24Þ
R 1 n −1 n R 1 n −1 n hs G dx x G dx −12V n R0 1 −11 c1 ¼ −6 0R 1 −11 n n G dx 1 0 0 G1 dx
ð25Þ
R1 G3 xn dxn −12V n 0R 1 n G3 dxn 0 G3 dx G2 dxn
ð26Þ
Integrating the mean film pressure yields the mean film force: Z F¼
A
ð27Þ
PBdx 0
Expressing in a dimensionless form gives Fn ¼
Z
2
Fhm0 2
μUA B
¼
1
pn dxn
ð28Þ
0
After performing integrations, the dimensionless mean film force can be obtained. Longitudinal : F n ¼ 6
Z
þ c1
Transverse : F n ¼ 6
1
0
Z
Z
Z
xn
n
hs dxn dxn G1
xn
0 xn
0
1
xn n n dx dx G1
1 dxn dxn G1
ð29Þ
G2 dxn dxn
0
0
Z
1
xn
0
1
Z
Z
Z 0
Z
0
þ 12V n þ c2
1
0
n
Z
ð17Þ
Hn g n ðH n ; N; LÞ
þ 12V
where δ denotes the shoulder parameter, Λ is the roughness parameter, and L represents the interacting parameter. Then the dimensionless stochastic non-Newtonian dynamic Reynolds equations can be expressed as ∂pn ∂ ∂EðH n Þ þ 12V n Longitudinal : n E g n ðH n ; N; LÞ ¼6 ð18Þ n ∂x ∂x ∂x Transverse :
g n ðH n ; N; LÞ
−Λ
G3 ðh ; N; L; ΛÞ ¼
2
x U hm h P hs n n ; pn ¼ m0 ; hs ¼ ¼ ð1−xn Þδ xn ¼ ; t n ¼ t; hm ¼ A A μUA hm0 hm0
Λ
Z
n
0
where P ¼ EðPÞ denotes the mean film pressure. In order to analyze the problem conveniently, the dimensionless variables and parameters are introduced as follows:
G2 dxn þ 12V n
0
−Λ
G2 ðh ; N; L; ΛÞ ¼
R1
ð12Þ
xn
ð20Þ
where
c2 ¼ −6 R01
Z
Z
ð9Þ
Transverse : H ¼ hðx; tÞ þ αðx; ξÞ
∂ ∂x
0
þc2
n
Transverse :
Z xn n n hs n x dx þ 12V n dxn G1 0 G1 Z xn 1 þc1 dxn 0 G1
xn
ð7Þ
where the function f represents the probability density distribution for the stochastic variable. Since most of the engineering rough surfaces are Gaussian in nature, a polynomial function is chosen to approach the Gaussian distribution [24]: ( 35ðλ2 −α2 Þ3 if−λ ≤α ≤ þ λ 32λ7 f ðαÞ ¼ ð8Þ 0 elsewhere
∂ Longitudinal : ∂x
Z
1
Z
0
Z
xn
G3 xn dxn dxn
0 xn
G3 dxn dxn
ð30Þ
0
Taking the film force F* under the steady state “0”, one can 2 obtain the dimensionless steady load capacity W n ¼ Whm0 =μUA2 B. ! Z 1 Z xn n hs Longitudinal : W n ¼ 6⋅ dxn dxn G1 0 0 0 ! Z 1 Z xn 1 n n þ ðc1 Þ0 ⋅ dx dx ð31Þ 0 0 G1 0
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
Z
n
Transverse : W ¼ 6⋅
1
Z
!
xn
n
n
G2 dx dx 0
Z þ ðc2 Þ0 ⋅
0
1 0
Z
xn
!
0
G3 dxn dxn
0
ð32Þ n
n
Snd ¼ −ð∂F n =∂hm0 Þ0
ð33Þ
Dnd ¼ −ð∂F n =∂V n Þ0
ð34Þ
As a result, one can derive the expressions for dimensionless dynamic coefficients: " !# Z 1 Z xn n ∂ hs n n dx dx Longitudinal : Snd ¼ −6 n ∂hm 0 0 G1 0 " !# Z 1 Z xn ∂ 1 n n −c1 dx dx n ∂hm 0 0 G1 0 ! ! Z 1 Z xn ∂c1 1 n n − ⋅ dx dx ð35Þ n ∂hm 0 0 G1 0 0 "
!# Z 1 Z xn ∂ n n G dx dx 2 n ∂hm 0 0 " !0# Z 1 Z xn ∂ −c2 ⋅ G3 dxn dxn n ∂hm 0 0 ! !0 Z 1 Z xn ∂c2 − ⋅ G3 dxn dxn n ∂hm 0 0 0 0
Transverse : Snd ¼ −6
Z
1
Z
Z
1 0
Z
xn
ð36Þ
xn
xn n n dx dx G 1 0 0 R 1 n −1 n Z 1 Z xn x G1 dx 1 þ 12 0R 1 −1 ⋅ dxn dxn n G 1 0 0 G dx 1 0
Transverse : Dnd ¼ −12
and (19) reduce to the Reynolds equation by Taylor and Dowson [1]: n n d dh n3 dp ¼6 n h ð39Þ n n dx dx dx
0
Performing the partial derivatives of F* with hm and V*, respectively, and then taking the resulting values under steady 3 state, one can obtain the stiffness coefficient Snd ¼ Sd hm0 =μUA2 B and 2 3 n the damping coefficient Dd ¼ Dd hm0 =μUA B:
Longitudinal : Dnd ¼ −12
153
ð37Þ
(b) Λ¼ 0, N ¼0 (or L¼ 0): dynamic plane slider bearings with a Newtonian lubricant. In this case, the present Reynolds equations reduce to the form by Lin et al. [3]: n n ∂ ∂h n3 ∂p ¼ 6 n þ 12V n h ð40Þ n n ∂x ∂x ∂x (c) Λ¼ 0: non-Newtonian dynamic plane slider bearings with a micropolar fluid. The Reynolds equations of the present derivation reduce to the form of Lin et al. [23]: n n n ∂ Nh ∂p ∂h n3 n n2 ¼ 6 n þ 12V n h þ 12L2 h −6NLh coth n n 2L ∂x ∂x ∂x ð41Þ
In this study, the coupled effects of surface roughness and micropolar fluids on dynamic characteristics of plane slider bearings are presented for Λ≠0, N≠0 and L≠0. Fig. 2 shows the steady load capacity W* as a function of the shoulder parameter δ for n different roughness parameter Λ under hm ¼0.5. The results for Λ¼0.2 and 0.4 describe the load capacity for the bearing with a micropolar fluid (N ¼0.5, L¼ 0.5) considering the effects of surface roughness. Bearing load capacities are observed to increase with the shoulder parameter until a maximum is obtained, and thereafter falls as the value of δ continues to increase. It is also observed that the influences of longitudinal roughness result in a decrease in W* as compared to the bearing with smooth surfaces, whereas the effects of transverse roughness yield an increase in the value of load capacity. Since the roughness pattern of transverse structure has the form of ridges and valleys running in the sliding direction, it tends to restrict the available flow and therefore diminish the fluid flow. As a result, an increase in the film pressure and the
G3 xn dxn dxn
0
R1 Z 1 Z xn G3 xn dxn þ 12 R0 1 ⋅ G3 dxn dxn n 0 0 0 G3 dx
ð38Þ
The dimensionless load capacity W* and the dimensionless dynamic coefficients Snd and Dnd can be calculated by the numerical integration method.
3. Results and discussion From the above analysis, the coupled effects of surface roughness and micropolar fluids on dynamic characteristics of bearings are influenced by three dimensionless parameters: the roughness parameter Λ ¼λ/hm0 defined in Eq. (17), the coupling parameter N ¼[k/(2μ+k)]1/2 defined in Eq. (5), and the interacting parameter L ¼l/hm0 ¼(γ/4μ)1/2/hm0 defined in Eq. (17). In addition, the shoulder parameter δ¼d/hm0 defined in Eq. (17) dominates the geometric effect on the slider bearing, and the squeezing velocity n V n ¼ dhm =dt n characterizes the effect of squeeze action of slider bearings. Some special cases can be recovered from the present study. (a) Λ ¼0, N ¼0 (or L¼0), V* ¼ 0: steady plane slider bearings with a Newtonian lubricant. Under this circumference, the dimensionless stochastic non-Newtonian dynamic Reynolds Eqs. (18)
Fig. 2. Load capacity W* as a function of the shoulder parameter δ for different Λ n under N ¼ 0.5, L ¼0.5 and hm ¼ 0.5.
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J.-R. Lin et al. / Tribology International 66 (2013) 150–156
integrated load capacity is expected. For longitudinal roughness structures, the form of ridges and valleys running perpendicular to the sliding direction tends to give an increase in lubricant flow; the effects of longitudinal roughness on the bearing pressure and load capacity are then reversed. Fig. 3 presents the dynamic stiffness coefficient Snd as a function of the shoulder parameter δ n for different Λ under N ¼0.5, L ¼0.5 and hm ¼0.5. Since the transverse roughness effects yield an increased load capacity and the longitudinal roughness effects result in a reduced load capacity, the derivative stiffness coefficients are similarly affected as compared to the case of smooth surfaces. It is also observed that
Fig. 3. Stiffness coefficient Snd as a function of the shoulder parameter δ for different n Λ under N ¼ 0.5, L ¼0.5 and hm ¼0.5.
Fig. 4. Damping coefficient Dnd as a function of the shoulder parameter δ for n different Λ under N ¼ 0.5, L ¼0.5 and hm ¼ 0.5.
increasing values of the roughness parameter increases the effects of surface roughness on the bearing stiffness. Fig. 4 shows the dynamic damping coefficient Dnd as a function of the shoulder n parameter δ for different Λ under N ¼0.5, L¼ 0.5 and hm ¼0.5. The values of the damping coefficient are observed to decrease with increasing values of the shoulder parameter. The effects of longitudinal roughness are observed to yield a decrease in the value of the damping coefficient as compared to the smooth bearing; however, the influences of transverse roughness predict an increase in the damping coefficient. It is also observed that the surface roughness effects on damping characteristics of the nonNewtonian lubricated slider bearings are further pronounced for larger values of the roughness parameter and small values of the shoulder parameter. In order to get an insight into the coupled influences of surface roughness and micropolar fluids on bearing characteristics, Figs. 5–7 present the effect of variation of Λ on the steady load capacity W* and the dynamic coefficients Snd and Dnd , respectively, n for different N under L¼ 0.5, hm ¼0.5 and δ¼0.6. The results of the curves for N ¼0 correspond to the slider bearing lubricated with a conventional Newtonian lubricant. And the results under the value of Λ¼0 describe the performances for bearing with smooth surfaces. The effects of non-Newtonian micropolar fluids (N ¼0.3 and 0.5) result in higher values of the load capacity and dynamic coefficients as compared to the bearing lubricated with a Newtonian fluid (N ¼ 0). In addition, increasing the values of the roughness parameter increases the effects of surface roughness on the bearing characteristics as compared to the smooth bearing (Λ¼ 0). It is also observed that the surfaces patterns of transverse roughness provide an improvement in the bearing performances, whereas the surfaces patterns of longitudinal roughness reduce the values of the steady load and dynamic coefficients. For engineering references, Tables 1–3 show the numerical values of the steady load capacity W* and dynamic coefficients Snd and Dnd of a wide bearing with rough surfaces and with non-Newtonian fluids for different Λ, N and L under the shoulder parameter δ¼1.2 n and the film height hm ¼1. It is also observed that the coupled effects of surface roughness and micropolar fluids on the steady load dynamic characteristics are further pronounced for the plane
Fig. 5. Effect of variation of Λ on the load capacity W* for different N under L ¼0.5, n hm ¼ 0.5 and δ ¼ 0.6.
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
155
Table 1 Load capacity W* of a wide non-Newtonian rough plane slider bearing for different n
Λ, N and L under the shoulder parameter δ ¼1.2 and the film height hm ¼ 1. Roughness parameter Λ
W* Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.1602 0.16 0.1592 0.158 0.1562
0.1753 0.175 0.1742 0.1728 0.1709
0.2105 0.2102 0.2092 0.2075 0.2052
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.1602 0.161 0.1634 0.1675 0.1737
0.1753 0.1762 0.1788 0.1833 0.1901
0.2105 0.2116 0.2147 0.2202 0.2284
Table 2 Stiffness coefficient Snd of a wide non-Newtonian rough plane slider bearing for n
different Λ, N and L under the shoulder parameter δ ¼ 1.2 and the film height hm ¼ 1. Roughness parameter Λ
n
Fig. 6. Effect of variation of Λ on the stiffness coefficient Sd for different N under n L ¼ 0.5, hm ¼ 0.5 and δ ¼ 0.6.
Snd Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.3196 0.3187 0.316 0.3116 0.3056
0.3505 0.3495 0.3465 0.3417 0.3351
0.4232 0.422 0.4184 0.4125 0.4045
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.3196 0.3224 0.3309 0.346 0.3694
0.3505 0.3535 0.3629 0.3795 0.4051
0.4232 0.4268 0.4381 0.4581 0.489
Table 3 Damping coefficient Dnd of a wide non-Newtonian rough plane slider bearing for n
different Λ, N and L under the shoulder parameter δ ¼ 1.2 and the film height hm ¼ 1. Roughness parameter Λ
Dnd Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.2671 0.2666 0.2654 0.2633 0.2604
0.2922 0.2917 0.2903 0.288 0.2848
0.3509 0.3503 0.3486 0.3458 0.419
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.2671 0.2679 0.2706 0.2751 0.2818
0.2922 0.2931 0.296 0.301 0.3083
0.3509 0.3521 0.3556 0.3616 0.3704
Fig. 7. Effect of variation of Λ on the damping coefficient Dnd for different N under n L ¼ 0.5, hm ¼ 0.5 and δ ¼ 0.6.
4. Conclusions
slider bearings operating with higher values of the coupling parameter N, and the interacting parameter L, and the roughness parameter Λ. In a recent research, the steady performances of rough inclined stepped composite bearings lubricated with micropolar fluids have been analyzed by Naduvinamani and Siddangouda [31]. Although their dimensionless definition of the roughness parameter is different from the present study, the tendencies of the roughness effects on the load capacity of inclined plane slider bearings lubricated with micropolar fluids are similar. These qualitative agreements provide a support to the present investigation.
The coupled effects of surface roughness and non-Newtonian lubricants on the dynamic performances of plane slider bearings have been investigated in the present paper. From the results obtained and discussed, conclusions can be drawn as follows. Based on the microcontinuum theory of micropolar fluids of Eringen [9] incorporating the surface roughness model of Christensen [24], a stochastic non-Newtonian dynamic Reynolds equation has been derived for wide plane slider bearings. It is found that the roughness structure of transverse patterns provides an increase in the steady load capacity and the dynamic coefficients as compared to the smooth bearing lubricated with micropolar fluids, whereas the longitudinal roughness structure yields a
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decrease in the load capacity and the stiffness and damping coefficients. On the whole, the coupled effects of surface roughness and micropolar fluids on the performance characteristics are more pronounced for the bearings operating with higher values of the non-Newtonian coupling parameter, the interacting parameter, and the surface roughness parameter. Some numerical values of the steady load capacity and dynamic coefficients are also included in tables for engineering references. References [1] Taylor CM, Dowson D. Turbulent lubrication theory-application to design. ASME Journal of Lubrication Technology 1974;96:36–46. [2] Capitao JW. Influence of turbulence on performance characteristics of the tilting pad thrust bearings. ASME Journal of Lubrication Technology 1974;96:110–7. [3] Lin JR, Lu RF, Ho MH, Lin MC. Dynamic characteristics of wide tapered-land slider bearings. Journal of the Chinese Institute of Engineers 2006;29:543–8. [4] Launder BE, Leschziner ML. Flow in finite-width, thrust bearings including inertia effects. ASME Journal of Lubrication Technology 1978;100:330–8. [5] Elrod HG, Anwar I, Colsher R. Transient lubricant films with inertia. ASME Journal of Lubrication Technology 1983;105:369–74. [6] Talmage G, Carpino M. A pseudospectral-finite difference analysis of an infinitely wide slider bearing with thermal and inertia effects. STLE Tribology Transactions 1997;40:251–8. [7] Shyu SH, Jeng YR, Chang CC. Load capacity for adiabatic infinitely wide plane slider bearings in turbulent thermodynamic regime. STLE Tribology Transactions 2004;47:396–401. [8] Scott W, Suntiwattana P. Effect of oil additives on the performance of a wet friction clutch material. Wear 1995;181-183:850–5. [9] Eringen AC. Theory of micropolar fluids. Journal Mathematics and Mechanics 1966;16:1–18. [10] Verma S, Kumar V, Gupta KD. Analysis of capillary compensated hydrostatic journal bearing operating with micropolar lubricant. Industrial Lubrication and Tribology 2011;63:192–202. [11] Nicodemus ER, Sharma SC. Orifice compensated multirecess hydrostatic/ hybrid journal bearing system of various geometric shapes of recess operating with micropolar lubricant. Tribology International 2011;44:284–96. [12] Ram N, Sharma SC. Analysis of orifice compensated non-recessed hole-entry hybrid journal bearing operating with micropolar lubricants. Tribology International 2012;52:132–43. [13] Gorla RSR, Nashery K. The contributions of microrotation of lubricant molecules in a thin film journal bearing. ACTA Mechanica 2002;158:185–98. [14] Das S, Guha SK, Chattopadhyay AK. On the conical whirl instability of hydrodynamic journal bearings lubricated with micropolar fluids. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 2001;215:431–9.
[15] Tandon PA, Jaggi S. A polar model for synovial fluid with reference to human joints. International Journal of Mechanical Science 1979;21:161–9. [16] Sinha P, Singh C. Micropolar squeeze films in porous hemispherical bearings. International Journal of Mechanical Science 1982;24:509–18. [17] Tandon PN, Chaurasia A. Microstructural effects on hip joint articulation. Applied Mathematical Modelling 1990;14:312–9. [18] Ramanaiah G, Dubey JN. Optimal slider bearing profile of a slider bearing lubricated with micropolar fluid. Wear 1977;42:1–7. [19] Isa M, Zaheeruddin K. Analysis of step bearings with a micropolar lubricant. Wear 1978;47:21–9. [20] Agrawal VK, Bhat SB. Porous pivoted slider bearings lubricated with a micropolar fluid. Wear 1980;61:1–8. [21] Bayada G, Boucha NB, Chambat M. New models in micropolar fluid and their application to lubrication. Mathematical Models and Methods in Applied Sciences 2005;15:343–74. [22] Naduvinamani NB, Marali GB. Dynamic Reynolds equation for micropolar fluids and the analysis of plane inclined slider bearings with squeezing effect. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 2007;221:823–9. [23] Lin JR, Chou TL, Liang LJ, Hung TC. Non-Newtonian dynamics characteristics of parabolic-film slider bearings: micropolar fluid model. Tribology International 2012;48:226–31. [24] Christensen H. Stochastic model for hydrodynamic lubrication of rough surfaces. Proceedings of the Institution of Mechanical Engineers 196970;184:1013–25. [25] Lin JR. Surface roughness effect on the dynamic stiffness and damping characteristics of compensated hydrostatic thrust bearings. International Journal of Machine Tools and Manufacture 2000;40:1671–89. [26] Turaga R, Sekhar AS, Majumdar BC. The effect of roughness parameter on the performance of hydrodynamic journal bearings with rough effects. Tribology International 1999;32:231–6. [27] Guha SK. Analysis of steady-state characteristics of misaligned hydrodynamic journal bearings with isotropic roughness effect. Tribology International 2000;22:1–12. [28] Christensen H, Tonder K. The hydrodynamic lubrication of rough bearing surfaces of finite width. ASME Journal of Lubrication Technology 1971;93:324–30. [29] Tonder K. Vibrations in spring-supported bearing pads due to non-conducting roughness. Wear 1999;232:256–61. [30] Andharia PI, Gupta JL, Deheri GM. Effect of surface roughness on hydrodynamic lubrication of slider bearings. STLE Tribology Transactions 2001;44:291–7. [31] Naduvinamani NB, Siddangouda A. On the performance of rough inclined stepped composite bearings with micropolar fluid. Journal of Marine Science and Technology – Taiwan 2010;18:233–42. [32] Siddangouda A. Surface roughness effects in porous inclined stepped composite bearings lubricated with micropolar fluid. Lubrication Science 2012;8:339–59.
Contents lists available at SciVerse ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Short Communication
Effects of surface roughness and non-Newtonian micropolar fluids on dynamic characteristics of wide plane slider bearings Jaw-Ren Lin a,n, Tzu-Chen Hung b, Tsu-Liang Chou a, Long-Jin Liang a a b
Department of Mechanical Engineering, Taoyuan Innovation Institute of Technology, Jhongli, Taiwan, ROC Department of Mechanical Engineering, National Taipei University of Technology, Taipei, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 16 February 2013 Received in revised form 5 May 2013 Accepted 7 May 2013 Available online 14 May 2013
Based on the Eringen's microcontinuum theory and the Christensen's stochastic theory, the combined effects of non-Newtonian rheology and surface roughness on the dynamic characteristics of slider bearings have been investigated. According to the results, the effects of transverse roughness provide an increase in the load capacity and dynamic coefficients as compared to the smooth bearing lubricated with micropolar fluids, whereas the influences of longitudinal roughness yield a reversed trend. The quantitative effects of rough surfaces and non-Newtonian fluids on bearing performances are more pronounced for higher values of the roughness parameter, the coupling parameter and the interacting parameter. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Surface roughness Non-Newtonian micropolar fluids Plane slider bearing Dynamic coefficients
1. Introduction Hydrodynamic slider bearings are commonly designed to sustain a transverse load in rotating machine systems. The performances of slider bearings have been investigated under various operating conditions, for example, the consideration of turbulent flows by Taylor and Dowson [1] and Capitao [2]; the inclusion of dynamic squeezing action by Lin et al. [3]; the inertia force effects by Launder and Leschziner [4] and Elrod et al. [5]; the thermal and inertia influences by Talmage and Carpino [6]; and the thermal turbulent effects by Shyu et al. [7]. All of these contributions focus on the use of a Newtonian lubricant. According to the experimental study of Scott and Suntiwattana [8], the use of additives provides a significant improvement in reducing the wear of wet clutch plates. Therefore, various types of additives are often added in lubricants to improve their performances. Since the conventional Newtonian theory is not adequate to describe these kinds of fluids containing substructures, a microcontinuum theory of micropolar fluids has been developed by Eringen [9]. The micropolar fluids can define spin inertia and microrotational effects generated by the local structures. According to the study of Eringen [9], micropolar fluids are important for fluids containing small particles, such as liquid crystals, animal bloods, polymetric suspensions and fluids blended additives. Therefore, the
n Corresponding author. No. 414, Sec. 3, Jhongshan E. Rd., Jhongli, 320, Taiwan, ROC. Tel.: +886 3 4361070x6212; fax: +886 3 4384670. E-mail address: [email protected] (J.-R. Lin).
0301-679X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.triboint.2013.05.004
theory of micropolar fluids has been applied to investigate the performance characteristics of various bearing systems, such as the capillary compensated hydrostatic bearings by Verma et al. [10], the orifice compensated hydrostatic/hybrid bearings by Nicodemus and Sharma [11], the orifice compensated nonrecessed hole-entry hybrid bearings by Ram and Sharma [12], the journal bearings by Gorla and Nashery [13] and Das et al. [14]; the squeeze film bearings by Tandon and Jaggi [15], Sinha and Singh [16] and Tandon and Chaurasia [17]; and the slider bearings by Ramanaiah and Dubey [18], Isa and Zaheeruddin [19], Agrawal and Bhat [20], Bayada et al. [21], Naduvinamani and Marali [22] and Lin et al. [23]. All of these contributions investigate the lubrication performances assuming that the bearing surfaces are perfectly smooth. However, bearing surfaces may be roughened owing to the manufacturing process. Therefore, the influences of surface roughness on bearing performances should be included. Applying the concept of a stochastic process, a modified Reynolds equation for rough bearing surfaces is derived by Christensen [24]. The longitudinal and transverse surface roughness effects on the operating characteristics of plane slider bearings are analyzed. It is reported that surface roughness may considerably influence the bearing characteristics. Based on the stochastic theory of Christensen [24], many contributions to the study of rough bearings have been presented, such as the investigations in the hydrostatic bearings by Lin [25]; the journal bearings by Turaga et al. [26] and Guha [27]; and the slider bearings by Christensen and Tonder [28], Tonder [29] and Andharia et al. [30]. According to their results, the consideration of surface roughness results in significant influences on the bearing characteristics. Recently, the steady performance of
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
Nomenclature A,B d
Dd,Dnd E F,F* h,h* h1 hm0 n hm ; hm n
hs ; hs H,H* k l L N
length and width of the bearing shoulder height 3 dynamic damping coefficient, Dnd ¼ Dd hm0 =μA3 B (dimensionless) expectancy operator 2 n dynamic film force, F n ¼ Fhm0 =μUA2 B ¼ F n ðhm ; V n Þ n n n n n smooth part of film profile, h ðx ; t Þ ¼ h=hm0 ¼ h þ hm (dimensionless) inlet film thickness steady outlet film thickness n outlet film thickness, hm ðt n Þ ¼ hm ðtÞ=hm0 (dimensionless) n mean film shape, hs ¼ hs =hm0 ¼ ð1−xn Þδ local film thickness, H* ¼H/hm0 ¼h*+α* (dimensionless) vortex viscosity coefficient of micropolar fluids characteristic material length, l ¼(γ/4μ)1/2 interacting parameter, L¼l/hm0 (dimensionless) coupling parameter, N ¼ [k/(2μ+k)]1/2 (dimensionless)
rough inclined stepped composite non-porous and porous slider bearings with micropolar fluids have been analyzed by Naduvinamani and Siddangouda [31] and Siddangouda [32], respectively. Since few studies consider the combined effects of surface roughness and non-Newtonian micropolar fluids on the dynamic characteristics of slider bearings, a further investigated is motivated. The aim of the present study is mainly concerned with the combined effects of surface roughness and micropolar fluids on the dynamic characteristics of plane slider bearings. Based on the microcontinuum theory of Eringen [9] incorporating the stochastic theory of Christensen [24], a stochastic non-Newtonian dynamic Reynolds equation is derived in the present study. Analytical expressions of the steady load capacity and the dynamic coefficients are obtained. The steady performance and the dynamic characteristics of the rough bearing lubricated with micropolar fluids are presented and discussed through the variation of the roughness parameter, the coupling parameter and the interacting parameter as compared to the smooth bearing lubricated with a Newtonian fluid. Some numerical values of plane bearing performances are also presented in tables for engineering references.
P P p* Sd,Snd t,t* u U V* W,W* x, y, z x* α γ δ λ Λ μ
151
local film pressure mean film pressure, P ¼ EðPÞ 2 mean film pressure, pn ¼ Phm0 =μUA (dimensionless) 3 mean stiffness coefficient, Snd ¼ Sd hm0 =μUA2 B (dimensionless) time, t* ¼Ut/A (dimensionless) velocity components in the x-direction sliding velocity of the lower part n squeezing velocity, V n ¼ dhm =dt n (dimensionless) 2 mean steady load capacity, W n ¼ Whm0 =μUA2 B (dimensionless) Cartesian coordinates horizontal coordinate, xn ¼ x=A (dimensionless) random part resulting from surface asperities spin gradient viscosity coefficient of micropolar fluids shoulder parameter, δ ¼ d=hm0 (dimensionless) half total range of random film thickness variable roughness parameter, Λ ¼ λ=hm0 (dimensionless) classical viscosity coefficients of Newtonian fluids
slider bearings lubricated with a micropolar fluid governing the local film pressure P can be written as derived by Lin et al. [23]: ∂ ∂P ∂H gðH; N; lÞ −6μUH ¼ 12μ ð3Þ ∂x ∂x ∂t where 2
gðH; N; lÞ ¼ H 3 þ 12l H−6NlH 2 coth½NH=ð2lÞ
ð4Þ
l ¼ ½γ=ð4μÞ1=2 ;
ð5Þ
N ¼ ½k=ð2μ þ kÞ1=2
In this equation, μ is the classical viscosity coefficient, γ is the spin gradient viscosity coefficient and k is the vortex viscosity coefficient responsible for micropolar fluids. In addition, l denotes the characteristic material length, N represents the dimensionless coupling parameter. Application of the stochastic approach of rough surfaces of Christensen [24] and taking the expected values for both sides of Eq. (3), one can obtain the stochastic non-Newtonian dynamic Reynolds equation: ∂ ∂P ∂EðHÞ E gðH; N; lÞ −6μUH ¼ 12μ ð6Þ ∂x ∂x ∂t
2. Analysis Fig. 1 shows the physical geometry of a wide inclined plane slider bearing with rough surfaces. The lubricant is taken to be a non-Newtonian incompressible micropolar fluid of Eringen [9]. The local film thickness H can be considered to be composed of two separated parts: H ¼ hðx; tÞ þ αðx; y; ξÞ
ð1Þ
where h(x,t) describes the nominal smooth part of the film profile depending on the horizontal coordinate x and the time t: hðx; tÞ ¼ hs ðxÞ þ hm ðtÞ ¼ ðA−xÞ
d þ hm ðtÞ A
ð2Þ
On the other hand, α(x,y,ξ) represents the random part as a result of the surface asperities measured from the nominal mean level, and ξ denotes a random variable describing a definite roughness arrangement. Assume that body forces and body couples are neglected and the usual thin film lubrication theory is applicable, the non-Newtonian dynamic Reynolds equation for
Fig. 1. Physical geometry of a wide rough plane slider bearing lubricated with a non-Newtonian micropolar fluid.
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J.-R. Lin et al. / Tribology International 66 (2013) 150–156
In this equation, E() denotes the expectancy operator defined by
Z
EðÞ ¼
∞ α ¼ −∞
ðÞf ðαÞdα
Longitudinal : pn ¼ 6
where λ describes the half total range of random film thickness variable. In the present study, it is mainly concerned with two types of roughness patterns: the longitudinal and transverse roughness structures. For one-dimensional longitudinal and transverse structures, the roughness patterns possess the form of narrow ridges and valleys running in the x and y-direction, respectively: Longitudinal : H ¼ hðx; tÞ þ αðy; ξÞ
Transverse : pn ¼ 6
ð10Þ
The stochastic non-Newtonian dynamic Reynolds equations are (
∂P E gðH; N; lÞ ∂x
(
1 ∂P E½g −1 ∂x
)
)
∂EðHÞ ∂EðHÞ þ 12μ ¼ 6μU ∂x ∂t
¼ 6μU
∂ E½H=gÞ ∂x E½1=g
ð11Þ
þ 12μ
∂EðHÞ ∂t
Z
n
G1 ðh ; N; L; ΛÞ ¼
ð13Þ n
h ¼ n
h α H n n n ¼ hs þ hm ; αn ¼ ; Hn ¼ ¼ h þ αn hm0 hm0 hm0 n
g ðH ; N; LÞ ¼
g (
n
f ðαn Þ ¼ hm0 f ¼
δ¼
¼H
3
hm0
2
n3
þ 12L H −6NLH coth NH n =ð2LÞ
n2 3
2
n
n2
35ðΛ −α Þ 32Λ7
if −Λ ≤αn ≤ þ Λ
0
elsewhere
d λ l ; Λ¼ ; L¼ hm0 hm0 hm0
ð14Þ ð15Þ
∂ ∂xn n
1 ∂pn E½g n−1 ∂xn
Λ
Z
Λ
¼6
∂ ∂xn
E½H n g n−1 þ 12V n n −1 E½g
1 g n ðH n ; N; LÞ
−Λ
ð16Þ
ð19Þ
where V n ¼ dhm =dt n denotes the squeezing velocity. Performing the integrations in the expectancy operator and applying the boundary conditions p*(x* ¼0) ¼0 and p*(x* ¼1) ¼0, one can obtain the dimensionless dynamic film pressure.
Z
xn
xn
G3 xn dxn
0
G3 dxn
0
ð21Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð22Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð23Þ
35ðΛ2 −αn2 Þ3 n dα 32Λ7
ð24Þ
R 1 n −1 n R 1 n −1 n hs G dx x G dx −12V n R0 1 −11 c1 ¼ −6 0R 1 −11 n n G dx 1 0 0 G1 dx
ð25Þ
R1 G3 xn dxn −12V n 0R 1 n G3 dxn 0 G3 dx G2 dxn
ð26Þ
Integrating the mean film pressure yields the mean film force: Z F¼
A
ð27Þ
PBdx 0
Expressing in a dimensionless form gives Fn ¼
Z
2
Fhm0 2
μUA B
¼
1
pn dxn
ð28Þ
0
After performing integrations, the dimensionless mean film force can be obtained. Longitudinal : F n ¼ 6
Z
þ c1
Transverse : F n ¼ 6
1
0
Z
Z
Z
xn
n
hs dxn dxn G1
xn
0 xn
0
1
xn n n dx dx G1
1 dxn dxn G1
ð29Þ
G2 dxn dxn
0
0
Z
1
xn
0
1
Z
Z
Z 0
Z
0
þ 12V n þ c2
1
0
n
Z
ð17Þ
Hn g n ðH n ; N; LÞ
þ 12V
where δ denotes the shoulder parameter, Λ is the roughness parameter, and L represents the interacting parameter. Then the dimensionless stochastic non-Newtonian dynamic Reynolds equations can be expressed as ∂pn ∂ ∂EðH n Þ þ 12V n Longitudinal : n E g n ðH n ; N; LÞ ¼6 ð18Þ n ∂x ∂x ∂x Transverse :
g n ðH n ; N; LÞ
−Λ
G3 ðh ; N; L; ΛÞ ¼
2
x U hm h P hs n n ; pn ¼ m0 ; hs ¼ ¼ ð1−xn Þδ xn ¼ ; t n ¼ t; hm ¼ A A μUA hm0 hm0
Λ
Z
n
0
where P ¼ EðPÞ denotes the mean film pressure. In order to analyze the problem conveniently, the dimensionless variables and parameters are introduced as follows:
G2 dxn þ 12V n
0
−Λ
G2 ðh ; N; L; ΛÞ ¼
R1
ð12Þ
xn
ð20Þ
where
c2 ¼ −6 R01
Z
Z
ð9Þ
Transverse : H ¼ hðx; tÞ þ αðx; ξÞ
∂ ∂x
0
þc2
n
Transverse :
Z xn n n hs n x dx þ 12V n dxn G1 0 G1 Z xn 1 þc1 dxn 0 G1
xn
ð7Þ
where the function f represents the probability density distribution for the stochastic variable. Since most of the engineering rough surfaces are Gaussian in nature, a polynomial function is chosen to approach the Gaussian distribution [24]: ( 35ðλ2 −α2 Þ3 if−λ ≤α ≤ þ λ 32λ7 f ðαÞ ¼ ð8Þ 0 elsewhere
∂ Longitudinal : ∂x
Z
1
Z
0
Z
xn
G3 xn dxn dxn
0 xn
G3 dxn dxn
ð30Þ
0
Taking the film force F* under the steady state “0”, one can 2 obtain the dimensionless steady load capacity W n ¼ Whm0 =μUA2 B. ! Z 1 Z xn n hs Longitudinal : W n ¼ 6⋅ dxn dxn G1 0 0 0 ! Z 1 Z xn 1 n n þ ðc1 Þ0 ⋅ dx dx ð31Þ 0 0 G1 0
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
Z
n
Transverse : W ¼ 6⋅
1
Z
!
xn
n
n
G2 dx dx 0
Z þ ðc2 Þ0 ⋅
0
1 0
Z
xn
!
0
G3 dxn dxn
0
ð32Þ n
n
Snd ¼ −ð∂F n =∂hm0 Þ0
ð33Þ
Dnd ¼ −ð∂F n =∂V n Þ0
ð34Þ
As a result, one can derive the expressions for dimensionless dynamic coefficients: " !# Z 1 Z xn n ∂ hs n n dx dx Longitudinal : Snd ¼ −6 n ∂hm 0 0 G1 0 " !# Z 1 Z xn ∂ 1 n n −c1 dx dx n ∂hm 0 0 G1 0 ! ! Z 1 Z xn ∂c1 1 n n − ⋅ dx dx ð35Þ n ∂hm 0 0 G1 0 0 "
!# Z 1 Z xn ∂ n n G dx dx 2 n ∂hm 0 0 " !0# Z 1 Z xn ∂ −c2 ⋅ G3 dxn dxn n ∂hm 0 0 ! !0 Z 1 Z xn ∂c2 − ⋅ G3 dxn dxn n ∂hm 0 0 0 0
Transverse : Snd ¼ −6
Z
1
Z
Z
1 0
Z
xn
ð36Þ
xn
xn n n dx dx G 1 0 0 R 1 n −1 n Z 1 Z xn x G1 dx 1 þ 12 0R 1 −1 ⋅ dxn dxn n G 1 0 0 G dx 1 0
Transverse : Dnd ¼ −12
and (19) reduce to the Reynolds equation by Taylor and Dowson [1]: n n d dh n3 dp ¼6 n h ð39Þ n n dx dx dx
0
Performing the partial derivatives of F* with hm and V*, respectively, and then taking the resulting values under steady 3 state, one can obtain the stiffness coefficient Snd ¼ Sd hm0 =μUA2 B and 2 3 n the damping coefficient Dd ¼ Dd hm0 =μUA B:
Longitudinal : Dnd ¼ −12
153
ð37Þ
(b) Λ¼ 0, N ¼0 (or L¼ 0): dynamic plane slider bearings with a Newtonian lubricant. In this case, the present Reynolds equations reduce to the form by Lin et al. [3]: n n ∂ ∂h n3 ∂p ¼ 6 n þ 12V n h ð40Þ n n ∂x ∂x ∂x (c) Λ¼ 0: non-Newtonian dynamic plane slider bearings with a micropolar fluid. The Reynolds equations of the present derivation reduce to the form of Lin et al. [23]: n n n ∂ Nh ∂p ∂h n3 n n2 ¼ 6 n þ 12V n h þ 12L2 h −6NLh coth n n 2L ∂x ∂x ∂x ð41Þ
In this study, the coupled effects of surface roughness and micropolar fluids on dynamic characteristics of plane slider bearings are presented for Λ≠0, N≠0 and L≠0. Fig. 2 shows the steady load capacity W* as a function of the shoulder parameter δ for n different roughness parameter Λ under hm ¼0.5. The results for Λ¼0.2 and 0.4 describe the load capacity for the bearing with a micropolar fluid (N ¼0.5, L¼ 0.5) considering the effects of surface roughness. Bearing load capacities are observed to increase with the shoulder parameter until a maximum is obtained, and thereafter falls as the value of δ continues to increase. It is also observed that the influences of longitudinal roughness result in a decrease in W* as compared to the bearing with smooth surfaces, whereas the effects of transverse roughness yield an increase in the value of load capacity. Since the roughness pattern of transverse structure has the form of ridges and valleys running in the sliding direction, it tends to restrict the available flow and therefore diminish the fluid flow. As a result, an increase in the film pressure and the
G3 xn dxn dxn
0
R1 Z 1 Z xn G3 xn dxn þ 12 R0 1 ⋅ G3 dxn dxn n 0 0 0 G3 dx
ð38Þ
The dimensionless load capacity W* and the dimensionless dynamic coefficients Snd and Dnd can be calculated by the numerical integration method.
3. Results and discussion From the above analysis, the coupled effects of surface roughness and micropolar fluids on dynamic characteristics of bearings are influenced by three dimensionless parameters: the roughness parameter Λ ¼λ/hm0 defined in Eq. (17), the coupling parameter N ¼[k/(2μ+k)]1/2 defined in Eq. (5), and the interacting parameter L ¼l/hm0 ¼(γ/4μ)1/2/hm0 defined in Eq. (17). In addition, the shoulder parameter δ¼d/hm0 defined in Eq. (17) dominates the geometric effect on the slider bearing, and the squeezing velocity n V n ¼ dhm =dt n characterizes the effect of squeeze action of slider bearings. Some special cases can be recovered from the present study. (a) Λ ¼0, N ¼0 (or L¼0), V* ¼ 0: steady plane slider bearings with a Newtonian lubricant. Under this circumference, the dimensionless stochastic non-Newtonian dynamic Reynolds Eqs. (18)
Fig. 2. Load capacity W* as a function of the shoulder parameter δ for different Λ n under N ¼ 0.5, L ¼0.5 and hm ¼ 0.5.
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J.-R. Lin et al. / Tribology International 66 (2013) 150–156
integrated load capacity is expected. For longitudinal roughness structures, the form of ridges and valleys running perpendicular to the sliding direction tends to give an increase in lubricant flow; the effects of longitudinal roughness on the bearing pressure and load capacity are then reversed. Fig. 3 presents the dynamic stiffness coefficient Snd as a function of the shoulder parameter δ n for different Λ under N ¼0.5, L ¼0.5 and hm ¼0.5. Since the transverse roughness effects yield an increased load capacity and the longitudinal roughness effects result in a reduced load capacity, the derivative stiffness coefficients are similarly affected as compared to the case of smooth surfaces. It is also observed that
Fig. 3. Stiffness coefficient Snd as a function of the shoulder parameter δ for different n Λ under N ¼ 0.5, L ¼0.5 and hm ¼0.5.
Fig. 4. Damping coefficient Dnd as a function of the shoulder parameter δ for n different Λ under N ¼ 0.5, L ¼0.5 and hm ¼ 0.5.
increasing values of the roughness parameter increases the effects of surface roughness on the bearing stiffness. Fig. 4 shows the dynamic damping coefficient Dnd as a function of the shoulder n parameter δ for different Λ under N ¼0.5, L¼ 0.5 and hm ¼0.5. The values of the damping coefficient are observed to decrease with increasing values of the shoulder parameter. The effects of longitudinal roughness are observed to yield a decrease in the value of the damping coefficient as compared to the smooth bearing; however, the influences of transverse roughness predict an increase in the damping coefficient. It is also observed that the surface roughness effects on damping characteristics of the nonNewtonian lubricated slider bearings are further pronounced for larger values of the roughness parameter and small values of the shoulder parameter. In order to get an insight into the coupled influences of surface roughness and micropolar fluids on bearing characteristics, Figs. 5–7 present the effect of variation of Λ on the steady load capacity W* and the dynamic coefficients Snd and Dnd , respectively, n for different N under L¼ 0.5, hm ¼0.5 and δ¼0.6. The results of the curves for N ¼0 correspond to the slider bearing lubricated with a conventional Newtonian lubricant. And the results under the value of Λ¼0 describe the performances for bearing with smooth surfaces. The effects of non-Newtonian micropolar fluids (N ¼0.3 and 0.5) result in higher values of the load capacity and dynamic coefficients as compared to the bearing lubricated with a Newtonian fluid (N ¼ 0). In addition, increasing the values of the roughness parameter increases the effects of surface roughness on the bearing characteristics as compared to the smooth bearing (Λ¼ 0). It is also observed that the surfaces patterns of transverse roughness provide an improvement in the bearing performances, whereas the surfaces patterns of longitudinal roughness reduce the values of the steady load and dynamic coefficients. For engineering references, Tables 1–3 show the numerical values of the steady load capacity W* and dynamic coefficients Snd and Dnd of a wide bearing with rough surfaces and with non-Newtonian fluids for different Λ, N and L under the shoulder parameter δ¼1.2 n and the film height hm ¼1. It is also observed that the coupled effects of surface roughness and micropolar fluids on the steady load dynamic characteristics are further pronounced for the plane
Fig. 5. Effect of variation of Λ on the load capacity W* for different N under L ¼0.5, n hm ¼ 0.5 and δ ¼ 0.6.
J.-R. Lin et al. / Tribology International 66 (2013) 150–156
155
Table 1 Load capacity W* of a wide non-Newtonian rough plane slider bearing for different n
Λ, N and L under the shoulder parameter δ ¼1.2 and the film height hm ¼ 1. Roughness parameter Λ
W* Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.1602 0.16 0.1592 0.158 0.1562
0.1753 0.175 0.1742 0.1728 0.1709
0.2105 0.2102 0.2092 0.2075 0.2052
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.1602 0.161 0.1634 0.1675 0.1737
0.1753 0.1762 0.1788 0.1833 0.1901
0.2105 0.2116 0.2147 0.2202 0.2284
Table 2 Stiffness coefficient Snd of a wide non-Newtonian rough plane slider bearing for n
different Λ, N and L under the shoulder parameter δ ¼ 1.2 and the film height hm ¼ 1. Roughness parameter Λ
n
Fig. 6. Effect of variation of Λ on the stiffness coefficient Sd for different N under n L ¼ 0.5, hm ¼ 0.5 and δ ¼ 0.6.
Snd Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.3196 0.3187 0.316 0.3116 0.3056
0.3505 0.3495 0.3465 0.3417 0.3351
0.4232 0.422 0.4184 0.4125 0.4045
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.3196 0.3224 0.3309 0.346 0.3694
0.3505 0.3535 0.3629 0.3795 0.4051
0.4232 0.4268 0.4381 0.4581 0.489
Table 3 Damping coefficient Dnd of a wide non-Newtonian rough plane slider bearing for n
different Λ, N and L under the shoulder parameter δ ¼ 1.2 and the film height hm ¼ 1. Roughness parameter Λ
Dnd Newtonian
N¼ 0.3, L ¼ 0.3
N ¼0.5, L ¼ 0.5
Smooth Λ ¼ 0.1 (Longitudinal) Λ ¼ 0.2 (Longitudinal) Λ ¼ 0.3 (Longitudinal) Λ ¼ 0.4 (Longitudinal)
0.2671 0.2666 0.2654 0.2633 0.2604
0.2922 0.2917 0.2903 0.288 0.2848
0.3509 0.3503 0.3486 0.3458 0.419
Smooth Λ ¼ 0.1 (Transverse) Λ ¼ 0.2 (Transverse) Λ ¼ 0.3 (Transverse) Λ ¼ 0.4 (Transverse)
0.2671 0.2679 0.2706 0.2751 0.2818
0.2922 0.2931 0.296 0.301 0.3083
0.3509 0.3521 0.3556 0.3616 0.3704
Fig. 7. Effect of variation of Λ on the damping coefficient Dnd for different N under n L ¼ 0.5, hm ¼ 0.5 and δ ¼ 0.6.
4. Conclusions
slider bearings operating with higher values of the coupling parameter N, and the interacting parameter L, and the roughness parameter Λ. In a recent research, the steady performances of rough inclined stepped composite bearings lubricated with micropolar fluids have been analyzed by Naduvinamani and Siddangouda [31]. Although their dimensionless definition of the roughness parameter is different from the present study, the tendencies of the roughness effects on the load capacity of inclined plane slider bearings lubricated with micropolar fluids are similar. These qualitative agreements provide a support to the present investigation.
The coupled effects of surface roughness and non-Newtonian lubricants on the dynamic performances of plane slider bearings have been investigated in the present paper. From the results obtained and discussed, conclusions can be drawn as follows. Based on the microcontinuum theory of micropolar fluids of Eringen [9] incorporating the surface roughness model of Christensen [24], a stochastic non-Newtonian dynamic Reynolds equation has been derived for wide plane slider bearings. It is found that the roughness structure of transverse patterns provides an increase in the steady load capacity and the dynamic coefficients as compared to the smooth bearing lubricated with micropolar fluids, whereas the longitudinal roughness structure yields a
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