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Effects of couple stresses in the cyclic squeeze films of finite partial journal bearings
Tribology International 34 (2001) 119–125 www.elsevier.com/locate/triboint
Effects of couple stresses in the cyclic squeeze films of finite partial journal bearings Jaw-Ren Lin *, Ching-Been Yang, Rong-Fang Lu Department of Mechanical Engineering, Nanya Institute of Technology, PO Box 267, Chung-Li 320, Taiwan, ROC Received 9 June 2000; received in revised form 11 November 2000; accepted 21 November 2000
Abstract Based upon the microcontinuum theory, the present paper is to theoretically study the pure squeeze-film behavior of a finite partial journal bearing with non-Newtonian couple-stress lubricants operating under a time-dependent cyclic load. To take into account the couple stress effects resulting from the lubricant blended with various additives, the modified Reynolds equation governing the film pressure is obtained from Stokes equations of motion. The film pressure is numerically solved by using the Conjugate Gradient Method. Bearing characteristics are then calculated from the nonlinear motion equation of the journal. According to the results obtained, the effects of couple stresses result in a decrease in the value of eccentricity of the journal center. The finite partial bearing with a couple stress fluid as the lubricant yields an increase in the minimum permissible clearance and provides a longer time to prevent the journal-bearing contact. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cyclic squeeze films; Finite partial bearings; Time-dependent loads; Nonlinear motion equation; Couple stress fluids
1. Introduction The squeeze film mechanism is of practical significance in many areas of engineering and is commonly observed in the bearings of automotive engines, aircraft engines, machine tools, turbomachinery, and skeletal joints. Conventionally, the prediction of squeeze film motion assumes that the lubricant behaves as a Newtonian viscous fluid. However, experimental results show that the addition of small amounts of long-chained additives to a Newtonian fluid minimizes the sensitivity of the lubricant to change in shear rate and provides beneficial effects on the load-carrying and frictional characteristics [1,2]. Moreover, a base oil blended with additives can stabilize the behavior of lubricants in elastohydrodynamic contacts and reduce friction and surface damage [3]. To describe the rheological behavior of this kind of non-Newtonian lubricant, many microcontinuum theories have been generated [4–6]. The Stokes theory [4] is the simplest generalization of the
* Corresponding author. Tel.: +886-3-4361070; fax: +886-34687031. E-mail address: [email protected] (J.-R. Lin).
classical theory of fluids which allows for polar effects such as the presence of couple stresses, body couples and non-symmetric tensors. This couple stress fluid is a special case of a non-Newtonian fluid and is intended to take account of particle-size effects. According to the Stokes microcontinuum theory, the field equations of an incompressible fluid with couple stresses are given by r
DV 1 ⫽⫺ⵜp⫹pB⫹ ⵜ⫻(rG)⫹(m⫺hⵜ2)ⵜ2V Dt 2
ⵜ·V⫽0
(1) (2)
where the vectors V, B, and G represent the velocity, body force per unit mass, and body couple per unit mass, respectively; r denotes the density, p is the pressure; m is the classical viscosity coefficient; and h is a new material constant responsible for the couple stress fluid property. Application of the couple stress model to biomechanic problems has been proposed in the study of peristaltic transport [7,8]. In lubrication fields many authors have successfully investigated the couple stress effects on different lubrication problems such as the externally pressurized bearings [9–11], the slider bearings [12], the rolling contact bearings [13–15], the squeeze film bearings [16–20]. In the previous rese-
0301-679X/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 0 0 ) 0 0 1 4 7 - X
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J.-R. Lin et al. / Tribology International 34 (2001) 119–125
Nomenclature B C e F F* G h h* h*min l l
*
L m* p p* R S t V Wd W(t) x,y,z z* e emax h q l m r t w ⵜ D Dt
body force vector radial clearance eccentricity film force nondimensional film force, F*=F/Wd body couple vector film thickness, Ch*=C⫺e cosq dimensionless film thickness, 1⫺ecosq dimensionless minimum permissible film clearance h 1/2 characteristic length of the additives, m l couple stress parameter, C length of the bearing mcw2 dimensionless mass of the journal, Wd film pressure pC2 dimensionless film pressure, mR2w radius of the journal mw R 2 Sommerfeld number, · Wd/2RL C time velocity vector amplitude of the dynamic applied load time-dependent load rectangular coordinates dimensionless coordinate in the z-direction e eccentricity ratio, C maximum eccentricity ratio material constant responsible for the couple stress property circumferential coordinate L length-to-diameter ratio, 2R classical viscosity coefficient density dimensionless response time, wt frequency of the oscillation gradient operator
冉冊
冉冊
material derivative
arches [19,20] the leading author has analyzed the squeeze film characteristics of a long partial journal bearing and a finite journal bearing subject to a steady load, respectively. However, the motion with a squeeze film mechanism often falls into the category that the bearings operate under dynamic loading conditions in which the applied load may vary in magnitude or direc-
tion with time. Consequently, the locus of the journal center will undergo fluctuations compatible with the variations in the applied load. Therefore a further study is needed. On the basis of the microcontinuum theory, this study is intended to predict the effects of couple stresses on the pure squeeze-film behavior of a finite bearing subject
J.-R. Lin et al. / Tribology International 34 (2001) 119–125
to a time-dependent load. The modified Reynolds equation is obtained by using the Stokes equations to account for the influence of couple stresses resulting from the lubricant blended with various additives. The film pressure is numerically solved and applied to derive the nonlinear motion equation of the journal. The dynamic squeeze-film characteristics (such as the velocity of the journal center, the locus of the journal center, and the minimum permissible film clearance) for the bearing lubricated with a couple stress fluid will be compared to the Newtonian-lubricant case.
2. Analysis Consider the physical configuration of a pure-squeezing partial journal bearing with length L shown as in Fig. 1. The journal of radius R is approaching the bearing surface with a velocity dh/dt. The lubricant in the system is taken to be an incompressible non-Newtonian couple stress fluid. It is assumed that the body forces and body couples are absent, the fluid film is thin as compared to the radius of journal, and fluid inertia is small as compared to the viscous shear. The velocity components of the lubricant are solved from Eq. (1). Substituting the expressions of velocity components into continuity Eq. (2) and integrating with respect to z, one has the dimensionless modified Reynolds equation [19,20].
冋
册
冋
册
∂ ∗ ∗ ∗ ∂p∗ 1 ∂ ∂p∗ f (h ,l ) ⫹ 2 ∗ f ∗(h∗,l∗) ∗ ∂q ∂q 4l ∂z ∂z de ⫽12 cos q dt where f* is a function of h* and l*
(3)
121
f ∗(h∗,l∗)⫽h∗3⫺12l∗2h∗⫹24l∗3 tanh
冉 冊 h∗ 2l∗
(4)
In the equation q and z* represent the dimensionless circumferential and axial coordinates respectively, t the dimensionless response time, p* the dimensionless squeeze film pressure, l the length-to-diameter ratio, e the eccentricity ratio, h* the dimensionless film thickness, and l* the couple stress parameter. The boundary conditions for the film pressure are 1 p∗⫽0 at z∗⫽⫹ 2
(5a)
1 p∗⫽0 at z∗⫽⫺ 2
(5b)
p p∗⫽0 at q⫽⫺ 2
(5c)
p p∗⫽0 at q⫽⫹ 2
(5d)
The modified Reynolds equation is now solved numerically by using a finite difference scheme. The film domain is divided by grid spacing [20]. The mesh for the film extent has 36 intervals in the circumferential direction, and 20 intervals across the bearing length. In finite increment format the terms in Eqs. (5a)–(5d) are given by
冋 册 冋 冋 册
∗ ∗ ∂ ∗∂p∗ p∗i+1,j −pi,j p∗i,j −pi−1,j 1 f ⫽ fi+1,j ⫺f ∗i−1,j 2 ∂q ∂q ⌬q 2 ⌬q ⌬q
冋
册
1 ∂ ∗∂p∗ p∗ −p∗i,j 1 ∗ 1 i,j+1 f ⫽ f i,j+ 2 4l2∂z∗ ∂z∗ 4l2⌬z∗ ⌬z∗
册
(6a) (6b)
p∗i,j −p∗i,j−1 2 ⌬z∗
∗ 1 ⫺f i,j−
de de 12 cos q⫽12 cos qi dt dt
(6c)
Substituting these expressions in the modified Reynolds equation one has ∗ A0p∗i,j ⫹A1p∗i+1,j ⫹A2p∗i−1,j ⫹A3pi,j+1 ⫹A4p∗i,j−1⫽Bi,j
(7)
where A0, A1, A2, A3, A4, and Bi,j are given by ∗1 ∗1 ∗ 1 ∗ 1 A0⫽4l2r∗2(f i+ ,j ⫹f i− ,j )⫹f i,j+ ⫹f i,j−
(8a)
A1⫽⫺4l2r∗2f ∗i+1,j
(8b)
A2⫽⫺4l2r∗2f ∗i−1,j
(8c)
∗ 1 A3⫽⫺f i,j+
(8d)
2
2
2
2
Fig. 1.
Physical configuration of a finite partial journal bearing.
2
2
2
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J.-R. Lin et al. / Tribology International 34 (2001) 119–125
A4⫽⫺f ∗i,j−1
(8e)
2
Bi,j ⫽⫺48l ⌬z
∗2
2
de cos qi dt
⌬z∗ r∗ ⫽ ⌬q
(8f) (8g)
∗ ri,j ⫽A0pi,j ⫹A1p∗i+1,j ⫹A2p∗i−1,j ⫹A3p∗i,j+1⫹A4p∗i,j−1⫺Bi,j
(9)
Then the pressure values have converged according to the following convergence criteria.
冋 冘冘
册
k+1 rk+1 i,j r j,i |
|
rki,j rkj,i|
1/2
⬍10−3
(10)
With the film pressure obtained, the hydrodynamic squeeze film force can be evaluated.
3. Dynamic squeeze film characteristics By integrating the film pressure acting on the journal shaft one obtains the dynamic squeeze film force.
冕 冕 z=L/2
q=p/2
z=0
q=−p/2
F⫽⫺2R
p cos q·dq dz
(11)
The dynamic equation of the journal shaft is then W(t)⫺F⫽mC
d 2e dt2
(12)
where W(t) represents the dynamic applied load and m is the mass of the journal. Assume that the time-dependent load is of the sinusoidal oscillating form: W(t)⫽Wd sin wt
(13)
where the amplitude Wd is smaller than the absolute value of the dynamic squeeze film force. Introduce the Sommerfeld number S and the dimensionless mass of the journal m* S⫽
冉 冊
de d 2e sin t S ⫽ ⫹ g e, dt2 m∗ m∗ dt
(15)
where g(e,de/dt) is
Eq. (7) with the boundary conditions Eqs. (5a)–(5d) is numerically solved for the pressure using the Conjugate Gradient Method (CGM) [21–23]. Based on a three-term recurrence, the CGM is an unconstrained optimization technique in which the search directions are conjugate. The finite termination property implies that this method is guaranteed to terminate after a finite number of steps. Let the residual vector ri,j be
|
As a result, Eq. (12) results in the time-eccentricity relationship:
冉冊
mw R 2 ∗ mCw2 , m ⫽ Wd/2RL C Wd
(14)
冉 冊
g e,
冘冘 m
n
de ⫽⌬q⌬z∗ p∗i,j cos qi dt i⫽0j⫽0
(16)
Eq. (15) is a highly nonlinear equation of second order, which describes the locus of the journal center. With the initial conditions e(0) and de(0)/dt given, the velocity of the journal center de/dt and the eccentricity ratio e(t) can be numerically evaluated by using the fourth-order Runge–Kutta method [24].
4. Results and discussion According to the Stokes microcontinuum theory, h is a material constant responsible for the couple stress property. Since the dimension of m is that of viscosity and the dimension of h is that of momentum, the ratio (h/m) has dimensions of length squared, and the dimension of l=(h/m)1/2 is of length. Consequently, this length l may be considered as some property, depending upon the molecular dimensions of the additives in a Newtonian lubricant. With the aid of the definition l*=l/C, this couple stress parameter characterizes the couple stress effects on the system. As the value of l* tends to zero, the modified Reynolds Eq. (4) reduces to the Newtonianlubricant case [25,26]. Therefore, it is expected that the couple stress effects would be prominent when the value of l* is large. On the other hand, the material constant, h, could be determined by some experiments as discussed by Stokes [4]. The couple stress parameter is then determined. To consist with the previous studies [20], bearing characteristics are presented for the values of l* =0.1 and 0.2. As an example, the initial conditions and the bearing data are: e(0)⫽0.4, de(0)/dt⫽0.00001, L⫽0.2 m, R⫽0.1 m, m∗ ⫽5, C⫽0.00001 m, m⫽0.015 Pa·sec, Wd⫽40⫻103 N, w ⫽200 rad/sec. Then: l⫽1, C/R⫽0.001, S⫽3. With the data given above, the dynamic behavior of the squeeze-film bearing can be predicted.
J.-R. Lin et al. / Tribology International 34 (2001) 119–125
Fig. 2. Dimensionless fluid film force F* as a function of dimensionless response time t.
Fig. 2 shows the dimensionless fluid film force F* as a function of dimensionless response time t resulting from the bearing lubricated with a Newtonian lubricant and couple stress fluids. It is observed that the presence of couple stresses yields an increase in the absolute value of F* in the time intervals (0,p/2) and (3p/2,2p) as well as a decrease in the time interval (p/2,3p/2). Fig. 3 depicts the velocity of the journal center de/dt versus dimensionless response time t for different couple stress parameter l*. With the initial conditions specified, the journal center picks up the velocity as the time increases. It is found that the bearing lubricated with couple stress
Fig. 3. Velocity of the journal center de/dt versus dimensionless response time t.
123
fluids decreases the velocity of the journal center. For the bearing subject to a sinusoidal oscillating load, the journal is pushed away from the initial position within the time intervals (0,p/2) and (p,3p/2) and is pulled back towards the original state within the time intervals (p/2,p) and (3p/2,p). Since the value of the film forces stands for the ability to sustain the load, the effects of couple stresses provide more resistances for the journal center from being pushed away from the initial eccentricity. Therefore, the bearing lubricated with couple stress fluids acts with a slower moving velocity of the journal center. Fig. 4 illustrates the locus of the journal center versus dimensionless response time t with Sommerfeld number S=1 and S=3. It is observed for S=3 that the journal displacement is reduced by the use of couple stress fluids. Since the couple stress lubricants cause the fluid to offer more restances to the journal motion such that the velocity of the journal center is reduced, it allows for smaller e. It is also observed that the effects of couple stresses are more pronounced with a smaller value of Sommerfeld number (S=1). To provide a safe operation of the squeezing-bearing system, an important factor in designing the squeeze film bearings is the minimum permissible clearance. The relationship between the minimum permissible clearance h∗min and the maxmimum eccentricity ratio emax is h∗min⫽1⫺emax
(17)
With a maxmimum eccentricity ratio corresponding to a minimum permissible clearance, a smaller displacement of the journal center would yield a larger h∗min. The variation of h∗min versus Sommerfeld number S for different
Fig. 4. Locus of the journal center e versus dimensionless response time t.
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J.-R. Lin et al. / Tribology International 34 (2001) 119–125
couple stress parameter l* is presented in Fig. 5. It is observed that the bearing with couple stress fluids results in larger values of h∗min. Since the couple stress lubricant provides a reduction in the velocity of the journal motion such that it allows for a smaller eccentricity, a larger permissible clearance is thereby achieved. It is also observed that the quantitative effects of couple stresses on the minimum permissible clearance are more pronounced with small values of Sommerfeld number. The squeeze-film mechanism with couple stress fluids subject to a constant applied load has been analyzed for a long partial journal bearing [19] and a finite journal bearing [20], respectively. Comparing the dynamic characteristics with the previous results is practically difficult, since in the present study the definition of dimensionless film pressure (p*=pC2/mR2w) is different from the steady case (p*=pC2/mR2(de/dt)). However, when the applied load is constant, the nonlinear second-order equation of motion, Eq. (12) reduces to the steady squeeze-film case. The previous studies are thus recovered.
5. Conclusions On the basis of the Stokes microcontinuum theory, the effects of couple stresses on the dynamic squeeze film characteristics of finite partial journal bearings are presented. The dynamic film pressure is numerically solved from the modified Reynolds equation, which takes into account the couple stress effects resulting from the lubricant blended with various additives. Considering the bearing subject to a time-dependent sinusoidal load,
Fig. 5. Variation of the dimensionless minimum permissible clearance h∗min versus Sommerfeld number S.
the nonlinear motion equation of the journal is then analyzed. According to the results disscussed, conclusions can be drawn as follows. Compared with the Newtonian-lubricant case, the effects of couple stresses provide more resistance to the dynamic squeeze film motion as well as a reduction in the journal-center velocity, and thereby allow for a larger minimum permissible clearance. By the use of a couple stress fluid as the lubricant, the bearing provides a longer time to prevent the journal-bearing contact and results in longer bearing life. When the applied load is constant, the steady squeeze-film characteristics in the previous studies [19,20] are recovered.
References [1] Oliver DR. Load enhancement effects due to polymer thickening in a short model journal bearing. J Non-Newtonian Fluid Mechanics 1988;30:185–96. [2] Scott W, Suntiwattana P. Effect of oil additives on the performance of a wet friction clutch material. Wear 1995;181/183: 850–5. [3] Spikes HA. The behaviour of lubricants in contacts: current understanding and future possibilities. J Proc Instn Mech Engrs 1994;28:3–15. [4] Stokes VK. Couple stresses in fluids. The Physics of Fluids 1966;9:1709–15. [5] Ariman TT, Sylvester ND. Microcontinum fluid mechanics. A review. Int J Engr Sci 1973;11:905–30. [6] Ariman TT, Sylvester ND. Applications of microcontinum fluid mechanics. Int J Engr Sci 1974;12:273–93. [7] Srivastava LM. Peristaltic transport of a couple-stress fluid. Rheol Acta 1986;25:638–41. [8] Shehawey EFE, Mekheimer KS. Couple-stresses in peristaltic transport of fluids. J Phys D Appl Phys 1994;27:1163–70. [9] Ramanaiah G, Sarkar P. Squeeze films and thrust bearings lubricated by fluids with couple stress. Wear 1978;48:309–16. [10] Gupta RS, Sharma LG. Analysis of couple stress lubricant in hydrostatic thrust bearings. Wear 1988;48:257–69. [11] Lin JR. Static and dynamic behaviour of externally pressurized circular step thrust bearings lubricated with couple stress fluids. Tribology Int 1999;32:207–16. [12] Ramanaiah G. Slider bearings lubricated by fluids with couple stress. Wear 1979;52:27–36. [13] Sinha P, Singh C. Couple stresses in the lubrication of rolling contact bearings considering cavitation. Wear 1981;67:85–91. [14] Bujurke NM, Naduvinami NG. The lubrication of lightly cylinders in combined rolling, sliding and normal motion with couple stress fluid. Int J Mech Sci 1990;32:969–79. [15] Das NC. Elastohydrodynamic lubrication theory of line contacts: couple stress fluid model. STLE Trib Trans 1997;40(2):353–9. [16] Ramanaiah G. Squeeze films between finite plates lubricated by fluids with couple stress. Wear 1979;54:315–20. [17] Bujurke NM, Jayaraman G. The influence of couple stresses in squeeze films. Int J Mech Sci 1982;24:369–76. [18] Bujurke NM, Patil HP. An analysis of squeeze film action in porous layered bearings. Wear 1991;145:385–97. [19] Lin JR. Squeeze film characteristics of long partial journal bearings lubricated with couple stress fluids. Tribology Int 1997;30(1):53–8. [20] Lin JR. Squeeze film characteristics of finite journal bearings: couple stress fluid model. Tribology Int 1998;31(4):201–7.
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[21] Mitchell AR, Griffiths DA. The finite difference method in partial differential equations. New York: John Wiley and Sons, 1981. [22] Golub GH, Van Loan CF. Matrix computation. Baltimore, MD: The Johns Hopkins University Press, 1985. [23] Strikwerda JC. Finite difference schemes and partial differential equations. California: Wadsworth, 1989.
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[24] Burden RL, Faires JD, Reynolds AC. Numerical analysis. Boston: The Southeast Book Company, 1981. [25] Pinkus O, Sternlicht B. Theory of hydrodynamic lubrication. New York: McGraw-Hill, 1961. [26] Hamrock BJ. Fundamental of fluid film lubrication. New York: McGraw-Hill, 1994.
Effects of couple stresses in the cyclic squeeze films of finite partial journal bearings Jaw-Ren Lin *, Ching-Been Yang, Rong-Fang Lu Department of Mechanical Engineering, Nanya Institute of Technology, PO Box 267, Chung-Li 320, Taiwan, ROC Received 9 June 2000; received in revised form 11 November 2000; accepted 21 November 2000
Abstract Based upon the microcontinuum theory, the present paper is to theoretically study the pure squeeze-film behavior of a finite partial journal bearing with non-Newtonian couple-stress lubricants operating under a time-dependent cyclic load. To take into account the couple stress effects resulting from the lubricant blended with various additives, the modified Reynolds equation governing the film pressure is obtained from Stokes equations of motion. The film pressure is numerically solved by using the Conjugate Gradient Method. Bearing characteristics are then calculated from the nonlinear motion equation of the journal. According to the results obtained, the effects of couple stresses result in a decrease in the value of eccentricity of the journal center. The finite partial bearing with a couple stress fluid as the lubricant yields an increase in the minimum permissible clearance and provides a longer time to prevent the journal-bearing contact. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cyclic squeeze films; Finite partial bearings; Time-dependent loads; Nonlinear motion equation; Couple stress fluids
1. Introduction The squeeze film mechanism is of practical significance in many areas of engineering and is commonly observed in the bearings of automotive engines, aircraft engines, machine tools, turbomachinery, and skeletal joints. Conventionally, the prediction of squeeze film motion assumes that the lubricant behaves as a Newtonian viscous fluid. However, experimental results show that the addition of small amounts of long-chained additives to a Newtonian fluid minimizes the sensitivity of the lubricant to change in shear rate and provides beneficial effects on the load-carrying and frictional characteristics [1,2]. Moreover, a base oil blended with additives can stabilize the behavior of lubricants in elastohydrodynamic contacts and reduce friction and surface damage [3]. To describe the rheological behavior of this kind of non-Newtonian lubricant, many microcontinuum theories have been generated [4–6]. The Stokes theory [4] is the simplest generalization of the
* Corresponding author. Tel.: +886-3-4361070; fax: +886-34687031. E-mail address: [email protected] (J.-R. Lin).
classical theory of fluids which allows for polar effects such as the presence of couple stresses, body couples and non-symmetric tensors. This couple stress fluid is a special case of a non-Newtonian fluid and is intended to take account of particle-size effects. According to the Stokes microcontinuum theory, the field equations of an incompressible fluid with couple stresses are given by r
DV 1 ⫽⫺ⵜp⫹pB⫹ ⵜ⫻(rG)⫹(m⫺hⵜ2)ⵜ2V Dt 2
ⵜ·V⫽0
(1) (2)
where the vectors V, B, and G represent the velocity, body force per unit mass, and body couple per unit mass, respectively; r denotes the density, p is the pressure; m is the classical viscosity coefficient; and h is a new material constant responsible for the couple stress fluid property. Application of the couple stress model to biomechanic problems has been proposed in the study of peristaltic transport [7,8]. In lubrication fields many authors have successfully investigated the couple stress effects on different lubrication problems such as the externally pressurized bearings [9–11], the slider bearings [12], the rolling contact bearings [13–15], the squeeze film bearings [16–20]. In the previous rese-
0301-679X/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 0 0 ) 0 0 1 4 7 - X
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J.-R. Lin et al. / Tribology International 34 (2001) 119–125
Nomenclature B C e F F* G h h* h*min l l
*
L m* p p* R S t V Wd W(t) x,y,z z* e emax h q l m r t w ⵜ D Dt
body force vector radial clearance eccentricity film force nondimensional film force, F*=F/Wd body couple vector film thickness, Ch*=C⫺e cosq dimensionless film thickness, 1⫺ecosq dimensionless minimum permissible film clearance h 1/2 characteristic length of the additives, m l couple stress parameter, C length of the bearing mcw2 dimensionless mass of the journal, Wd film pressure pC2 dimensionless film pressure, mR2w radius of the journal mw R 2 Sommerfeld number, · Wd/2RL C time velocity vector amplitude of the dynamic applied load time-dependent load rectangular coordinates dimensionless coordinate in the z-direction e eccentricity ratio, C maximum eccentricity ratio material constant responsible for the couple stress property circumferential coordinate L length-to-diameter ratio, 2R classical viscosity coefficient density dimensionless response time, wt frequency of the oscillation gradient operator
冉冊
冉冊
material derivative
arches [19,20] the leading author has analyzed the squeeze film characteristics of a long partial journal bearing and a finite journal bearing subject to a steady load, respectively. However, the motion with a squeeze film mechanism often falls into the category that the bearings operate under dynamic loading conditions in which the applied load may vary in magnitude or direc-
tion with time. Consequently, the locus of the journal center will undergo fluctuations compatible with the variations in the applied load. Therefore a further study is needed. On the basis of the microcontinuum theory, this study is intended to predict the effects of couple stresses on the pure squeeze-film behavior of a finite bearing subject
J.-R. Lin et al. / Tribology International 34 (2001) 119–125
to a time-dependent load. The modified Reynolds equation is obtained by using the Stokes equations to account for the influence of couple stresses resulting from the lubricant blended with various additives. The film pressure is numerically solved and applied to derive the nonlinear motion equation of the journal. The dynamic squeeze-film characteristics (such as the velocity of the journal center, the locus of the journal center, and the minimum permissible film clearance) for the bearing lubricated with a couple stress fluid will be compared to the Newtonian-lubricant case.
2. Analysis Consider the physical configuration of a pure-squeezing partial journal bearing with length L shown as in Fig. 1. The journal of radius R is approaching the bearing surface with a velocity dh/dt. The lubricant in the system is taken to be an incompressible non-Newtonian couple stress fluid. It is assumed that the body forces and body couples are absent, the fluid film is thin as compared to the radius of journal, and fluid inertia is small as compared to the viscous shear. The velocity components of the lubricant are solved from Eq. (1). Substituting the expressions of velocity components into continuity Eq. (2) and integrating with respect to z, one has the dimensionless modified Reynolds equation [19,20].
冋
册
冋
册
∂ ∗ ∗ ∗ ∂p∗ 1 ∂ ∂p∗ f (h ,l ) ⫹ 2 ∗ f ∗(h∗,l∗) ∗ ∂q ∂q 4l ∂z ∂z de ⫽12 cos q dt where f* is a function of h* and l*
(3)
121
f ∗(h∗,l∗)⫽h∗3⫺12l∗2h∗⫹24l∗3 tanh
冉 冊 h∗ 2l∗
(4)
In the equation q and z* represent the dimensionless circumferential and axial coordinates respectively, t the dimensionless response time, p* the dimensionless squeeze film pressure, l the length-to-diameter ratio, e the eccentricity ratio, h* the dimensionless film thickness, and l* the couple stress parameter. The boundary conditions for the film pressure are 1 p∗⫽0 at z∗⫽⫹ 2
(5a)
1 p∗⫽0 at z∗⫽⫺ 2
(5b)
p p∗⫽0 at q⫽⫺ 2
(5c)
p p∗⫽0 at q⫽⫹ 2
(5d)
The modified Reynolds equation is now solved numerically by using a finite difference scheme. The film domain is divided by grid spacing [20]. The mesh for the film extent has 36 intervals in the circumferential direction, and 20 intervals across the bearing length. In finite increment format the terms in Eqs. (5a)–(5d) are given by
冋 册 冋 冋 册
∗ ∗ ∂ ∗∂p∗ p∗i+1,j −pi,j p∗i,j −pi−1,j 1 f ⫽ fi+1,j ⫺f ∗i−1,j 2 ∂q ∂q ⌬q 2 ⌬q ⌬q
冋
册
1 ∂ ∗∂p∗ p∗ −p∗i,j 1 ∗ 1 i,j+1 f ⫽ f i,j+ 2 4l2∂z∗ ∂z∗ 4l2⌬z∗ ⌬z∗
册
(6a) (6b)
p∗i,j −p∗i,j−1 2 ⌬z∗
∗ 1 ⫺f i,j−
de de 12 cos q⫽12 cos qi dt dt
(6c)
Substituting these expressions in the modified Reynolds equation one has ∗ A0p∗i,j ⫹A1p∗i+1,j ⫹A2p∗i−1,j ⫹A3pi,j+1 ⫹A4p∗i,j−1⫽Bi,j
(7)
where A0, A1, A2, A3, A4, and Bi,j are given by ∗1 ∗1 ∗ 1 ∗ 1 A0⫽4l2r∗2(f i+ ,j ⫹f i− ,j )⫹f i,j+ ⫹f i,j−
(8a)
A1⫽⫺4l2r∗2f ∗i+1,j
(8b)
A2⫽⫺4l2r∗2f ∗i−1,j
(8c)
∗ 1 A3⫽⫺f i,j+
(8d)
2
2
2
2
Fig. 1.
Physical configuration of a finite partial journal bearing.
2
2
2
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A4⫽⫺f ∗i,j−1
(8e)
2
Bi,j ⫽⫺48l ⌬z
∗2
2
de cos qi dt
⌬z∗ r∗ ⫽ ⌬q
(8f) (8g)
∗ ri,j ⫽A0pi,j ⫹A1p∗i+1,j ⫹A2p∗i−1,j ⫹A3p∗i,j+1⫹A4p∗i,j−1⫺Bi,j
(9)
Then the pressure values have converged according to the following convergence criteria.
冋 冘冘
册
k+1 rk+1 i,j r j,i |
|
rki,j rkj,i|
1/2
⬍10−3
(10)
With the film pressure obtained, the hydrodynamic squeeze film force can be evaluated.
3. Dynamic squeeze film characteristics By integrating the film pressure acting on the journal shaft one obtains the dynamic squeeze film force.
冕 冕 z=L/2
q=p/2
z=0
q=−p/2
F⫽⫺2R
p cos q·dq dz
(11)
The dynamic equation of the journal shaft is then W(t)⫺F⫽mC
d 2e dt2
(12)
where W(t) represents the dynamic applied load and m is the mass of the journal. Assume that the time-dependent load is of the sinusoidal oscillating form: W(t)⫽Wd sin wt
(13)
where the amplitude Wd is smaller than the absolute value of the dynamic squeeze film force. Introduce the Sommerfeld number S and the dimensionless mass of the journal m* S⫽
冉 冊
de d 2e sin t S ⫽ ⫹ g e, dt2 m∗ m∗ dt
(15)
where g(e,de/dt) is
Eq. (7) with the boundary conditions Eqs. (5a)–(5d) is numerically solved for the pressure using the Conjugate Gradient Method (CGM) [21–23]. Based on a three-term recurrence, the CGM is an unconstrained optimization technique in which the search directions are conjugate. The finite termination property implies that this method is guaranteed to terminate after a finite number of steps. Let the residual vector ri,j be
|
As a result, Eq. (12) results in the time-eccentricity relationship:
冉冊
mw R 2 ∗ mCw2 , m ⫽ Wd/2RL C Wd
(14)
冉 冊
g e,
冘冘 m
n
de ⫽⌬q⌬z∗ p∗i,j cos qi dt i⫽0j⫽0
(16)
Eq. (15) is a highly nonlinear equation of second order, which describes the locus of the journal center. With the initial conditions e(0) and de(0)/dt given, the velocity of the journal center de/dt and the eccentricity ratio e(t) can be numerically evaluated by using the fourth-order Runge–Kutta method [24].
4. Results and discussion According to the Stokes microcontinuum theory, h is a material constant responsible for the couple stress property. Since the dimension of m is that of viscosity and the dimension of h is that of momentum, the ratio (h/m) has dimensions of length squared, and the dimension of l=(h/m)1/2 is of length. Consequently, this length l may be considered as some property, depending upon the molecular dimensions of the additives in a Newtonian lubricant. With the aid of the definition l*=l/C, this couple stress parameter characterizes the couple stress effects on the system. As the value of l* tends to zero, the modified Reynolds Eq. (4) reduces to the Newtonianlubricant case [25,26]. Therefore, it is expected that the couple stress effects would be prominent when the value of l* is large. On the other hand, the material constant, h, could be determined by some experiments as discussed by Stokes [4]. The couple stress parameter is then determined. To consist with the previous studies [20], bearing characteristics are presented for the values of l* =0.1 and 0.2. As an example, the initial conditions and the bearing data are: e(0)⫽0.4, de(0)/dt⫽0.00001, L⫽0.2 m, R⫽0.1 m, m∗ ⫽5, C⫽0.00001 m, m⫽0.015 Pa·sec, Wd⫽40⫻103 N, w ⫽200 rad/sec. Then: l⫽1, C/R⫽0.001, S⫽3. With the data given above, the dynamic behavior of the squeeze-film bearing can be predicted.
J.-R. Lin et al. / Tribology International 34 (2001) 119–125
Fig. 2. Dimensionless fluid film force F* as a function of dimensionless response time t.
Fig. 2 shows the dimensionless fluid film force F* as a function of dimensionless response time t resulting from the bearing lubricated with a Newtonian lubricant and couple stress fluids. It is observed that the presence of couple stresses yields an increase in the absolute value of F* in the time intervals (0,p/2) and (3p/2,2p) as well as a decrease in the time interval (p/2,3p/2). Fig. 3 depicts the velocity of the journal center de/dt versus dimensionless response time t for different couple stress parameter l*. With the initial conditions specified, the journal center picks up the velocity as the time increases. It is found that the bearing lubricated with couple stress
Fig. 3. Velocity of the journal center de/dt versus dimensionless response time t.
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fluids decreases the velocity of the journal center. For the bearing subject to a sinusoidal oscillating load, the journal is pushed away from the initial position within the time intervals (0,p/2) and (p,3p/2) and is pulled back towards the original state within the time intervals (p/2,p) and (3p/2,p). Since the value of the film forces stands for the ability to sustain the load, the effects of couple stresses provide more resistances for the journal center from being pushed away from the initial eccentricity. Therefore, the bearing lubricated with couple stress fluids acts with a slower moving velocity of the journal center. Fig. 4 illustrates the locus of the journal center versus dimensionless response time t with Sommerfeld number S=1 and S=3. It is observed for S=3 that the journal displacement is reduced by the use of couple stress fluids. Since the couple stress lubricants cause the fluid to offer more restances to the journal motion such that the velocity of the journal center is reduced, it allows for smaller e. It is also observed that the effects of couple stresses are more pronounced with a smaller value of Sommerfeld number (S=1). To provide a safe operation of the squeezing-bearing system, an important factor in designing the squeeze film bearings is the minimum permissible clearance. The relationship between the minimum permissible clearance h∗min and the maxmimum eccentricity ratio emax is h∗min⫽1⫺emax
(17)
With a maxmimum eccentricity ratio corresponding to a minimum permissible clearance, a smaller displacement of the journal center would yield a larger h∗min. The variation of h∗min versus Sommerfeld number S for different
Fig. 4. Locus of the journal center e versus dimensionless response time t.
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couple stress parameter l* is presented in Fig. 5. It is observed that the bearing with couple stress fluids results in larger values of h∗min. Since the couple stress lubricant provides a reduction in the velocity of the journal motion such that it allows for a smaller eccentricity, a larger permissible clearance is thereby achieved. It is also observed that the quantitative effects of couple stresses on the minimum permissible clearance are more pronounced with small values of Sommerfeld number. The squeeze-film mechanism with couple stress fluids subject to a constant applied load has been analyzed for a long partial journal bearing [19] and a finite journal bearing [20], respectively. Comparing the dynamic characteristics with the previous results is practically difficult, since in the present study the definition of dimensionless film pressure (p*=pC2/mR2w) is different from the steady case (p*=pC2/mR2(de/dt)). However, when the applied load is constant, the nonlinear second-order equation of motion, Eq. (12) reduces to the steady squeeze-film case. The previous studies are thus recovered.
5. Conclusions On the basis of the Stokes microcontinuum theory, the effects of couple stresses on the dynamic squeeze film characteristics of finite partial journal bearings are presented. The dynamic film pressure is numerically solved from the modified Reynolds equation, which takes into account the couple stress effects resulting from the lubricant blended with various additives. Considering the bearing subject to a time-dependent sinusoidal load,
Fig. 5. Variation of the dimensionless minimum permissible clearance h∗min versus Sommerfeld number S.
the nonlinear motion equation of the journal is then analyzed. According to the results disscussed, conclusions can be drawn as follows. Compared with the Newtonian-lubricant case, the effects of couple stresses provide more resistance to the dynamic squeeze film motion as well as a reduction in the journal-center velocity, and thereby allow for a larger minimum permissible clearance. By the use of a couple stress fluid as the lubricant, the bearing provides a longer time to prevent the journal-bearing contact and results in longer bearing life. When the applied load is constant, the steady squeeze-film characteristics in the previous studies [19,20] are recovered.
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