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Hydrodynamic bearings modeling with alternate motion
Mechanics Research Communications 37 (2010) 590–597
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Hydrodynamic bearings modeling with alternate motion Antonio C. Bannwart a , Katia L. Cavalca b,∗ , Gregory B. Daniel b a b
Department of Petroleum Engineering, University of Campinas, Faculty of Mechanical Engineering, Postal Box 6122, 13083-970 Campinas, SP, Brazil Department of Mechanical Design, University of Campinas, Faculty of Mechanical Engineering, Postal Box 6122, 13083-970 Campinas, SP, Brazil
a r t i c l e
i n f o
Article history: Received 8 October 2009 Received in revised form 3 March 2010 Available online 17 July 2010 Keywords: Hydrodynamic bearings Alternate motion Lubrication
a b s t r a c t This paper presents an analysis of the lubrication problem in hydrodynamic bearings with alternate rotational motion. In contrast to conventional bearings, the shaft does not perform a complete revolution but inverts the direction of its rotation instead. The analysis follows the basic assumption of the Reynolds lubrication theory i.e. fluid inertia is neglected in the gap formed by the eccentric shaft with respect to the bearing. The differential equations of mass and momentum in the oil gap are integrated to give the velocity field, pressure distribution and hydrodynamic forces under simplified assumptions. To evaluate the model under known operational conditions a rotor dynamics approach is used. This allows evaluation of the lubrication conditions for this type of bearing. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The purpose of the present work is to develop an analytical mathematical model for the hydrodynamic lubrication of bearings having an alternate rotational motion. In this class of hydrodynamic bearings the shaft does not perform a complete turn inside the bearing. Instead, it keeps reversing the direction of the rotation whenever a certain angle is attained. Considering a bearing having a fixed shape of cylindrical type, the fundamental lubrication problem is formulated from a classical fluid mechanics approach in order to develop a mathematical model for the unsteady flow in the gap occupied by the lubricating fluid. Under simplifying assumptions, these equations are analytically integrated and compared with the classical Reynolds result. The importance of the lubrication of bearings which establish the connection between the connecting rod and slider is essential for the tribology concept in slider-crank mechanism (Flores et al., 2006) and constitutes the main motivation of this work. This is a preliminary investigation of the lubrication condition to be continued to future applications involving such mechanism. The mechanism of hydrodynamic lubrication was independently discovered and formulated by N.P. Petrov (1836–1920), B. Towers (1845–1904) and O. Reynolds (1842–1912). They perceived the process of lubrication as being not one of a mechanical interaction between two solid surfaces, but involving the dynamics of a fluid film separating them. The application of Reynolds theory (Reynolds, 1886) to represent the hydrodynamic lubrication phenomena led to reliable models for hydrostatic and hydrody-
∗ Corresponding author. Tel.: +55 1935213178; fax: +55 1932893722. E-mail address: [email protected] (K.L. Cavalca). 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.07.003
namic bearings to be established. In 1925, A. Stodola, professor at ETH Zürich proposed a bearing modeling as a flexible connecting element between rotors and their supporting structures. Ocvirk (1952) presented the Reynolds equation solution for short bearings, neglecting the circumferential pressure gradient. More recently, Hashish and Sankar (1984) considered nonlinear effects on the stiffness and damping of hydrodynamic bearings. Capone (1986, 1991) proposed a non-linear hydrodynamic force model to a rigid, symmetric and horizontal rotor. Childs (1994) proposed several models and their solutions to labyrinth seals and hydrodynamic bearings.Frene et al. (2006) presented the theories to obtain the characteristics of bearings operating in turbulent flow regime, and the effects of inertia forces in laminar and in turbulent flows were shown. Costa and Hutchings (2007) analyzed the influence of the textured steel surface on the hydrodynamic lubrication under reciprocating sliding conditions, using patterns of circular depressions, grooves and chevrons. A rotor dynamics approach based on static equilibrium between external loading and hydrodynamic force was presented by Cavalca and Cattaruzzi (2001) and applied in a simplified model by Gandara et al. (2005) to investigate the conditions that guarantee proper lubrication of the journal-bearing. The originality of this work is the application of the Reynolds’ lubrication theory in bearings with alternate motion shafts, taking into account the local acceleration of the fluid film due to the motion of the slider in the mathematical model. This research results aim to verify the lubrication condition of this kind of hydrodynamic bearing under controlled kinematics conditions for future application in the dynamic model of a slidercrank mechanism, and further, for applications in practical problems like piston-pin connection in internal combustion engines.
A.C. Bannwart et al. / Mechanics Research Communications 37 (2010) 590–597
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2. Mathematical model 2.1. System geometry The main geometrical features of the system under study are illustrated in Fig. 1. The slider and connecting rod are kept together by a gudgeon-pin which is supported by holes bored in the slider. The central portion of the pin passes through the connecting rod small-end eye. This hinged joint provides a direct transference of thrust from the slider to the connecting rod and at the same time permits the rod to pivot relatively to the cylinder axis with an alternating motion. The alternating movement of the connecting rod during operating conditions tends to squeeze the oil film alternately from one side of the pin to the other under semiboundary-lubrication conditions. At the big-end hole there is a connection with the crank, which is a journal-bearing of complete turn. In Fig. 1, R is the crank length, L is the connecting rod length, q is the angular displacement of the crank, A is the angular displacement of the connecting rod, YSL is the linear displacement of the slider and C is the distance between the translation axis of the slider and the rotation center of the crank. The formulation of the mathematical model for the journalbearing consists basically in using the knowledge acquired from classical fluid mechanics and its application to the lubrication of bearings with alternate motion (Hamrock et al., 2004; Gresham, 2004). Let us consider the lubricating oil completely filling the gap between the slider and connecting rod, and that the oil stock within the gap remains constant. This can be achieved by an adequate supply system whereby the oil is constantly pumped into the bearing, which works emerged in oil. Consider the journal-bearing sketched in Fig. 2. For small eccentricity, the thickness h of the oil film can be calculated as: h() = Cr (1 + ε cos )
Fig. 1. Slider, connecting rod and crank mechanism.
Fig. 2. Schematic view of the journal-bearing system and its local coordinates.
(1)
where Cr is the radial clearance, ε = e/Cr the eccentricity ratio and e is the eccentricity. In this figure Ob , Rb and Oj , Rj are the center and radius of bearing and shaft, respectively. The center line passing through Ob and Oj defines the origin of the -coordinate, with h = hmax at = 0 and h = hmin at = . Also, a local rectangular coordinate system (x, y) can be defined such that the x-axis is tangential to the bearing and the y-axis passes through the center of the bearing. 2.2. Journal-bearing kinematics Fig. 3 illustrates the kinematics of the journal-bearing system in the slider-crank mechanism. As the crank performs a complete xy (t) = −ωU0 a
As a first step in the hydrodynamic analysis an expression for xy (t) of the (x, y) coordinate system shown in the acceleration a Fig. 2 is required. This acceleration is the vector sum of the slider ˛ (t) of the cen p (t) and the tangential acceleration a acceleration a ter line. The slider acceleration can be derived from a conventional slider-crank kinematics (Doughty, 1988). In the most of practical application of a slider-crank mechanism, the ratio between the length of the crank and the conrod (R/L) is 0.25, which gives a conrod angle (˛(t)) boundary from −15◦ to +15◦ . This assumption leads to a minimum of cos ˛(t) close to 0.97 and it can be rounded to the unity for the sake of simplification of the kinematics model, allowing the focus on the lubrication model itself. In this case, the acceleration of the (x, y) system is:
L cos(ωt) ˆj − ωU0 sin(ωt) ˆ Rb
axy,r (, t) = −ωU0
L L cos(ωt) cos(˛ + ) ∼ cos(ωt) cos = −ωU0 Rb Rb
axy, (, t) = −ωU0
L L cos(ωt) sin(˛ + ) − ωU0 sin(ωt) ∼ cos(ωt) sin − ωU0 sin(ωt) = −ωU0 Rb Rb
(2)
2.3. Hydrodynamic lubrication problem turn with period T = 2/ω, the slider translates alternately along the Y-axis. At the same time, the angle ˛(t) between the vertical axis and the center line shown in Fig. 2 oscillates around the Y-axis. In the slider-crank mechanism, the constant rotation of the crank causes a translation acceleration to the slider. Thus, either the upper or the lower parts of the bearing surface can be closer to the journal’s, depending on which surface is performing work. The hydrodynamic problem is essentially the same whatever the situation considered.
The study of fluid mechanics with its physical properties and considerations leads to the proper understanding of the phenomena involved in lubrication. The present analysis is aimed at obtaining the instantaneous velocity and pressure distributions in the lubricating fluid, from which the hydrodynamic forces on the journal can be determined. Our approach is based on the analogy of the relative rotation between two cylindrical eccentric surfaces with the relative translation of two flat inclined surfaces. Stokes’
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A.C. Bannwart et al. / Mechanics Research Communications 37 (2010) 590–597
- shaft wall:
v (h, , t) = 0 (a);
vr (h, , t) = 0 (b)
(5)
- bearing wall:
v (0, , t) = U0 cos(ω t) (a);
vr (0, , t) = 0 (b)
(6)
The solution of Eqs. (3) and (4) subject to Eqs. (5) and (6) is searched in the form:
vr (y, , t) = Vr (y, ) eiωt (a);
v (y, , t) = V (y, ) eiωt (b);
p(, t) = P() eiωt (c)
(7)
where Vr , V and P are complex amplitudes. Replacing Eq. (7) and their derivatives into Eqs. (3) and (4): −
Fig. 3. Journal-bearing kinematics.
second problem treatment (White, 1991) was then employed for the formulation in the complex domain. Consider the journal-bearing described in Fig. 2 where the journal shaft is rotating alternately in a very small gap filled with a lubricating fluid (usually oil). The basic assumptions originally proposed by Reynolds in his lubrication theory remain valid when the shaft performs an alternate motion. Accordingly, we assume that the gap is thin in comparison to the circumferential length of the bearing, h 2 Rb . As a consequence, the radial velocity component is small in comparison with the tangential component, i.e. Vr V . We also adopt the assumption of negligible inertia (Stokes flow), so that (V Cr2 )/( Rb ) 1. The fluid can be considered incompressible and possible cavitation effects are neglected. However, the flow is periodic with at time scale related to the crank period (T = 2/ω), thus the steady state flow assumption can not be adopted anymore. At this regard the goal of the present analysis is to investigate how the lubricating film flow is affected by the crank angular velocity. The assumptions of ∂/∂z = 0 and Vz = 0 is taken into account as in an infinite bearing. Thus, the flow is two-dimensional and can be represented in polar coordinates. According to Fig. 2, the radial coordinate r can be related to the y-axis through y = Rb − r, thus y = 0 at the bearing wall and y = h() at the journal wall. The angular coordinate is related to the x-axis through x = Rb since the gap is very thin. The (x, y) system is the usual coordinate frame for translational bearings. The problem consists in determining the tangential and radial velocity components defined by v (y, , t) and vr (y, , t), respectively, as well as the pressure field p(y, , t) in the lubricating fluid. From the above mentioned assumptions, the mass conservation equation can be written as: −
r-direction :
0=
V (h, ) = 0 (a);
Vr (h, ) = 0 (b);
1 ∂p ∂y
and
-direction :
∂v 1 ∂p ∂2 v + axy, (, t) = − + 2 Rb ∂ ∂t ∂y
V (0, ) = U0 (c); (9)
where Pt = P() + ωRb U0
L Rb
cos + i
(10)
The solution of Eq. (8) subject to Eqs. (9a)–(9d) is:
V (y, ) =
iω
U0 sinh (h() − y)
sinh h()
dPt iω Rb d
+
iω
⎧ ⎫ ⎨ sinh (h() − y) iω + sinh y iω ⎬ × −1 (11) ⎩ ⎭ sinh h() iω
Vr (y, ) =
U0 dh Rb d
iω
cosh y
−1
sinh2 h()
+
iω
iω Rb2
∂ ∂
dP
t
d
B(y, ) (12)
where
iω
− cosh (h() − y)
iω
+ cosh h()
−1
iω
+ cosh y
iω
−y
iω
sinh h()
(13)
The remaining condition, Eq. (9b), can be applied to Eq. (12) to give Eq. (14).
⎧ ⎡ ⎤⎫ ⎬ 3/2 d ⎨ dP 2 cosh h() iω −1 iω t ⎣ ⎦ h() − iω Rb d ⎩ Rb d ⎭ sinh h() iω
(4)
where and are the fluid’s specific mass and kinematics viscosity, respectively and axy, (, t) is given by Eq. (2). The boundary conditions are:
(8)
Vr (0, ) = 0 (d)
(3)
The Navier–Stokes equations greatly simplify thanks to the assumption of Stokes flow in the thin gap. However, a new term must be included to account for the non-inertial character of the (x, y) local coordinate system which moves with the central line. The momentum equations with negligible gravity effects become:
(b)
subject to:
B(y, ) =
∂vr 1 ∂v =0 + Rb ∂ ∂y
∂2 V 1 dPt iω − V = Rb d ∂ y2
1 ∂V ∂Vr + = 0 (a); Rb ∂ ∂y
=
iω
U0 Rdhd cosh h() b
sinh2 h()
iω
−1
(14)
A.C. Bannwart et al. / Mechanics Research Communications 37 (2010) 590–597
Further integration of Eq. (14) with the P(0) = P(2) = Po gives the pressure distribution as: P() = Po + ωRb U0
+ Rb U0
iω
L Rb
0
(1 − cos ) − i
tanh
h()
h() 2 iω
iω
condition
− K1 C2r
− 2 tanh
iω
d h( ) 2
(15)
iω
where
⎧ ⎫ h() iω ⎬ tanh 2 ⎨ 2 − 1 d 0 iω ⎩ h() iω −2tanh h() ⎭ 2 K1 = 2 Cr iω d 0 2 h() h()
iω −2 tanh
2
d Rb d
h3 ()
dPt,0 Rb d
= 6 U0
dh Rb d
Table 1 Some typical values for a journal-bearing system. Parameter and units
Symbol
Value
Bearing radius (m) Journal shaft radius (m) Bearing width (m) Radial clearance (gap) (m) Crank angular velocity (rad/s) Absolute viscosity (Pa s) Mass density (kg/m3 ) Crank length (m) Connecting rod length (m)
Rb Rj w Cr ω R L
10 × 10−3 10 × 10−3 15 × 10−3 20 × 10−6 <600 0.0117 887.8 0.040 0.00; 0.10; 0.20
force on the journal shaft (y = h) will be:
(16)
2
Fx (t) = wRj 0
iω
At the limiting case when ω → 0 the several imaginary parts vanish and Eqs. (14)–(16) and Eq. (14) reduces to:
593
Fy (t) = wRj 0
2
∂v −psin + cos ∂y
∂v −p cos + sin ∂y
(17)
It can be seen that, for dx = Rb d, Eq. (17) corresponds to the classical equation of the Reynolds lubrication theory. These results will be used for comparison. The solution for the velocity and pressure amplitudes in the alternate bearing, Eqs. (11)–(16), is thus dependent upon three dimensionless parameters, namely: C = Cr (ω/), L/Rb and . The square of the first parameter represents the Reynolds number of the alternate motion; for C 1 the velocity profiles approach those of classical lubrication theory and the alternate bearing behaves as a conventional bearing. The L/Rb ratio is associated with the slider acceleration and exerts a further influence on the pressure distribution around the shaft. The eccentricity ratio ε can assume any value (between 0 and 1) required to keep the shaft completely lubricated. The final periodic solutions for the tangential velocity component and pressure are given in Eq. (7), whereby it can be seen that both the real and imaginary parts of the amplitudes V (y, ) and P() play an important role for determining the real part of v (y, , t) and p(, t). For small eccentricity the local instantaneous hydrodynamic
d = Fx ei(ωt+ x ) y=h
(18)
d = Fy ei(ωt+ y ) y=h
(19)
where x and y are phase angles and the directions (x, y) are indicated in Fig. 2. 3. Numerical results In the following simulations we assume given values for the three independent dimensionless parameters C , L/Rb and ε. The lubricating conditions such as pressure distribution, velocity profile and hydrodynamic force are calculated accordingly. The influence of critical parameters such as crank rotation, slider acceleration, and eccentricity are discussed. The range of the above dimensionless parameters can be determined from the reference values listed in Table 1. The simulations are strictly valid for a suitable choice of the velocity U0 satisfying (U0 Cr2 )/(Rb ) 1. Fig. 4 illustrates the amplitude and phase of the dimensionless pressure P∗ = (P − Po )/(( U0 Rb )/Cr2 ) as a function of for some typical values of parameter C at L/Rb = 0 and ε = 0.5. As can be noted in Fig. 4(a), in the 0 < C < 0.25 range the journalbearing behavior tends to that described by the classical Reynolds
Fig. 4. Dimensionless pressure around the shaft (L/Rb = 0 and ε = 0.5): (a) amplitude and (b) phase.
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Fig. 5. Dimensionless pressure around the shaft ( C = 0.25 and L/Rb = 0) (a) amplitude and (b) phase.
equations. The imaginary component depends on C and its influence in the pressure distribution is smaller in comparison with the real component. However, Fig. 4b shows that the phase inversion of 90◦ causes a very small pressure distribution in the upper lobe of the cylindrical bearing. The effects of the eccentricity ratio on pressure distribution amplitude and phase are illustrated in Fig. 5(a) and (b), respectively, for ε = 0, 0.25, 0.5 and 0.75 at L/Rb = 0 (this assumption is used only to vanish the local acceleration in the theoretical analysis) and C = 0.25. As can be observed the position of the pressure peak approaches * = as the eccentricity ratio increases. When ε = 1 the oil film is broken, causing a direct contact of journal and bearing at = . Fig. 6 shows the angular position * of the pressure peak as a function of the eccentricity for several L/Rb and C = 0.25. For L/Rb = 0, * is nearly /2 for ε = 0; the limit condition * → as ε → 1 is noted. The slider acceleration can be observed to have a significant influence at low eccentricity. Fig. 7 shows the amplitude and phase of the dimensionless velocity profiles (V (Y, ))/U0 for several angular positions around the journal shaft, where Y = y/h. Both amplitude and phase components are important in view of Eq. (7). For the selected value
Fig. 6. Angular position of the pressure peak for several L/Rb and C = 0.25.
Fig. 7. Dimensionless velocity for several eccentricities and C = 0.25: (a) amplitude and (b) phase.
A.C. Bannwart et al. / Mechanics Research Communications 37 (2010) 590–597
595
Fig. 8. Dimensionless force Fx∗ for L/Rb = 0, 10 and 20 at C = 0.25: (a) amplitude and (b) phase.
Fig. 9. Dimensionless force Fy∗ for L/Rb = 0, 10 and 20 at C = 0.25: (a) amplitude and (b) phase.
C = 0.25, the velocity amplitudes become nearly identical to the classical lubrication theory in the bearing coordinate /4 < < . However, the velocity amplitudes have negatives values from = 0 (0.6 < Y < 1.0) to = /4 (0.65 < Y < 1.0), respectively. This effect is probably caused by the alternating motion of the bearing and it depends on the rotational velocity of the crank (ω). Figs. 8 and 9 show the amplitude and phase of the force component Fx ∗ = (Fx )/((wRj U0 Rb )/(Cr2 )) and Fy ∗ = (Fy )/((wRj U0 Rb )/(Cr2 )) on the journal shaft as functions of the eccentricity for L/Rb = 0, 10 and 20 at C = 0.25. The effect of slider acceleration appears on the forces amplitudes Fx * and Fy * as seen in Figs. 8(a) and 9(a), where the effect of shear has been omitted. In this case, the increasing of L/Rb increases the forces amplitudes due to the slider acceleration, which increases the pressure peak. Notice that the force amplitude Fx * is negative and Fy * is positive (Figs. 8(b) and 9(b), respectively). To verify the time-dependence of the hydrodynamic forces behavior, some numerical results were obtained for one lap of the crank (q = ωt, where ω is constant). In this simulation, the force actuating on the bearing is proportional to the slider acceleration (kinematics condition). Figs. 10 and 11 show the amplitude and phase of the instantaneous hydrodynamic force component Fx * and Fy * for one crank lap,
considering L/Rb = 20 and C = 0.095 (ω = 3000 rpm). Furthermore, the initial angle to start the motion of the crank is 90◦ . As can be seen in Figs. 10(a) and 11(a), the hydrodynamic forces Fx * and Fy * present quite similar behavior. The hydrodynamic forces describe a continuous curve, with a parabolic tendency, from 90◦ to 270◦ and, analogously, from 270◦ to 450◦ (90◦ ). However, the behavior is different due to the inversion of the slider acceleration direction (the loading, in this case). This inversion is the cause of the discontinuity presented at approximately 270◦ in all graphs (Figs. 10 and 11). Figs. 12 and 13 show the behavior of the eccentricity ratio for one crank lap. According to Fig. 12, the eccentricity ratio reduces when the crank angle goes from 90◦ to 180◦ because the increasing of the conrod angular velocity gradient is greater than the loading gradient. Sequentially, the eccentricity ratio increases when the crank angle approaches 270◦ , because the angular velocity of the conrod vanishes. At 270◦ , the slider acceleration inversion occurs and the shaft (conrod) finds a new equilibrium position at the opposite quadrant of the bearing. Consequently, there is a discontinuity in the simulation results due to the fact that only the kinematics analysis is carried out at this point of the analysis (without inertial effects).
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Fig. 10. Instantaneous dimensionless force Fx∗ for one crank lap, L/Rb = 20 and C = 0.095 (a) amplitude and (b) phase.
Fig. 11. Instantaneous dimensionless force Fy∗ for one crank lap, L/Rb = 20 and C = 0.095 (a) amplitude and (b) phase.
Fig. 12. Eccentricity ratio behavior for one crank lap. Fig. 13. Orbit of the shaft for one crank lap.
A.C. Bannwart et al. / Mechanics Research Communications 37 (2010) 590–597
Fig. 13 shows the orbit of the shaft (conrod end) in the bearing for one lap of the crank. Initially, the shaft is located in the fourth (right lower) quadrant of the bearing. However, after the inversion of the slider acceleration, the shaft goes to the second (upper left) quadrant of the bearing.
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Acknowledgements The authors thank to Fapesp, CNPq and CAPES for supporting different parts of this research. References
4. Conclusions This work was aimed at the hydrodynamic analysis of lubrication for bearings with alternating rotational motion shafts, as those present in slider-crank mechanisms. The approach was based on an analogy between the relative rotation between two cylindrical eccentric surfaces and the relative translation of two flat inclined surfaces and lubrication theory assumptions were adopted. A simplified kinematics of the slider-crank mechanism was included. The problem was formulated as to determine the force on the conrod end inside the slider hole for a given eccentricity, geometry, crank rotation, slider acceleration and fluid properties. An analytical solution was obtained, which revealed three dimensionless parameters governing the hydrodynamic force, namely C , L/Rb and ε. The first parameter represents the effect of the Reynolds number related to the crank rotation, which seems to be relevant for C > 1. This situation is outside the range focused in this work, but might be important for other journal-bearing sizes and rotations. The second effect represents the (linearized) slider acceleration effect, which was shown to be important in the present numerical simulations, especially because it generates a force in the direction of the slider motion. The effect of eccentricity was observed to be strong, confirming the results of previous works. However, in a real problem, the designer wants to ensure that the shaft will be properly lubricated (i.e. ε < 1). Thus, investigations are suggested to explore the dynamics of the mechanism under loading and its connection with the conrod end position inside the slider hole, in order to reach the lubrication map of this kinetic pair.
Capone, G., 1986. Moti orbilali di un rotore rigido simmetrico su cuscinetti cilindrici lubrificati. La Meccanica Italiana (199), 37–46. Capone, G., 1991. Descrizione analitica del campo di forze fluidodinamico nei cuscinetti cilindrici lubrificati. L’Energia Elettrica (3), 105–110. Cavalca, K.L., Cattaruzzi, E.F., 2001. A CAE tool dedicated to the design of hidrodynamic bearings. The International Journal of Mechanical Engineering Education, Manchester, UK 29 (4), 361–380. Childs, D., 1994. Turbomachinery Rotordynamics—Phenomena Modelling and Analysis. John Wiley & Sons, New York. Costa, H.L., Hutchings, I.M., 2007. Hydrodynamic lubrication of textured steel surfaces under reciprocating sliding conditions. Tribology International 40, 1227–1238. Doughty, S., 1988. Mechanics of Machines. John Wiley & Sons, Inc. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S., 2006. A study on dynamics of mechanical systems including joints with clearance and lubrication. Mechanism and Machine Theory 41, 247–261. Frene, J., Arghir, M., Constantinescu, V., 2006. Combined thin-film and Navier–Stokes analysis in high Reynolds number lubrication. Tribology International 39 (8), 734–747. Gandara, I., Bannwart, A.C., Cavalca, K.L., 2005. Hydrodynamic lubrication applied to bearings with oscillating motion. In: Proceedings of COBEM-2005, Ouro PretoBrazil, pp. 1–10. Gresham, R.M., 2004. What’s tribology. Tribology and Lubrication Technology 60 (11), 26–28. Hamrock, B.J., Schmid, S.R., Jacobson, B.O., 2004. Fundamentals of Fluid Film Lubrication, 2nd ed. Marcel Dekker, New York, Basel. Hashish, E., Sankar, T.S., 1984. Finite element and modal analyses of rotor-bearing systems under stochastic loading conditions. Journal of Vibration, Acoustics, Stress and Reliability in Design [s.l] 106 (4), 80–89. Ocvirk, F.W., 1952. Short-Bearing Approximation for Full Journal Bearings, National Advisory Committee for Aeronautics (NACA), Technical Report TN, 2808. Reynolds, O., 1886. On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments including an experimental determination of the viscosity of olive oil. Philos. Trans. R. Soc. Lond. Ser. A 177 (Part 1), 157–234. White, F., 1991. Viscous Fluid Flow. McGraw-Hill, New York.
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Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Hydrodynamic bearings modeling with alternate motion Antonio C. Bannwart a , Katia L. Cavalca b,∗ , Gregory B. Daniel b a b
Department of Petroleum Engineering, University of Campinas, Faculty of Mechanical Engineering, Postal Box 6122, 13083-970 Campinas, SP, Brazil Department of Mechanical Design, University of Campinas, Faculty of Mechanical Engineering, Postal Box 6122, 13083-970 Campinas, SP, Brazil
a r t i c l e
i n f o
Article history: Received 8 October 2009 Received in revised form 3 March 2010 Available online 17 July 2010 Keywords: Hydrodynamic bearings Alternate motion Lubrication
a b s t r a c t This paper presents an analysis of the lubrication problem in hydrodynamic bearings with alternate rotational motion. In contrast to conventional bearings, the shaft does not perform a complete revolution but inverts the direction of its rotation instead. The analysis follows the basic assumption of the Reynolds lubrication theory i.e. fluid inertia is neglected in the gap formed by the eccentric shaft with respect to the bearing. The differential equations of mass and momentum in the oil gap are integrated to give the velocity field, pressure distribution and hydrodynamic forces under simplified assumptions. To evaluate the model under known operational conditions a rotor dynamics approach is used. This allows evaluation of the lubrication conditions for this type of bearing. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The purpose of the present work is to develop an analytical mathematical model for the hydrodynamic lubrication of bearings having an alternate rotational motion. In this class of hydrodynamic bearings the shaft does not perform a complete turn inside the bearing. Instead, it keeps reversing the direction of the rotation whenever a certain angle is attained. Considering a bearing having a fixed shape of cylindrical type, the fundamental lubrication problem is formulated from a classical fluid mechanics approach in order to develop a mathematical model for the unsteady flow in the gap occupied by the lubricating fluid. Under simplifying assumptions, these equations are analytically integrated and compared with the classical Reynolds result. The importance of the lubrication of bearings which establish the connection between the connecting rod and slider is essential for the tribology concept in slider-crank mechanism (Flores et al., 2006) and constitutes the main motivation of this work. This is a preliminary investigation of the lubrication condition to be continued to future applications involving such mechanism. The mechanism of hydrodynamic lubrication was independently discovered and formulated by N.P. Petrov (1836–1920), B. Towers (1845–1904) and O. Reynolds (1842–1912). They perceived the process of lubrication as being not one of a mechanical interaction between two solid surfaces, but involving the dynamics of a fluid film separating them. The application of Reynolds theory (Reynolds, 1886) to represent the hydrodynamic lubrication phenomena led to reliable models for hydrostatic and hydrody-
∗ Corresponding author. Tel.: +55 1935213178; fax: +55 1932893722. E-mail address: [email protected] (K.L. Cavalca). 0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.07.003
namic bearings to be established. In 1925, A. Stodola, professor at ETH Zürich proposed a bearing modeling as a flexible connecting element between rotors and their supporting structures. Ocvirk (1952) presented the Reynolds equation solution for short bearings, neglecting the circumferential pressure gradient. More recently, Hashish and Sankar (1984) considered nonlinear effects on the stiffness and damping of hydrodynamic bearings. Capone (1986, 1991) proposed a non-linear hydrodynamic force model to a rigid, symmetric and horizontal rotor. Childs (1994) proposed several models and their solutions to labyrinth seals and hydrodynamic bearings.Frene et al. (2006) presented the theories to obtain the characteristics of bearings operating in turbulent flow regime, and the effects of inertia forces in laminar and in turbulent flows were shown. Costa and Hutchings (2007) analyzed the influence of the textured steel surface on the hydrodynamic lubrication under reciprocating sliding conditions, using patterns of circular depressions, grooves and chevrons. A rotor dynamics approach based on static equilibrium between external loading and hydrodynamic force was presented by Cavalca and Cattaruzzi (2001) and applied in a simplified model by Gandara et al. (2005) to investigate the conditions that guarantee proper lubrication of the journal-bearing. The originality of this work is the application of the Reynolds’ lubrication theory in bearings with alternate motion shafts, taking into account the local acceleration of the fluid film due to the motion of the slider in the mathematical model. This research results aim to verify the lubrication condition of this kind of hydrodynamic bearing under controlled kinematics conditions for future application in the dynamic model of a slidercrank mechanism, and further, for applications in practical problems like piston-pin connection in internal combustion engines.
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2. Mathematical model 2.1. System geometry The main geometrical features of the system under study are illustrated in Fig. 1. The slider and connecting rod are kept together by a gudgeon-pin which is supported by holes bored in the slider. The central portion of the pin passes through the connecting rod small-end eye. This hinged joint provides a direct transference of thrust from the slider to the connecting rod and at the same time permits the rod to pivot relatively to the cylinder axis with an alternating motion. The alternating movement of the connecting rod during operating conditions tends to squeeze the oil film alternately from one side of the pin to the other under semiboundary-lubrication conditions. At the big-end hole there is a connection with the crank, which is a journal-bearing of complete turn. In Fig. 1, R is the crank length, L is the connecting rod length, q is the angular displacement of the crank, A is the angular displacement of the connecting rod, YSL is the linear displacement of the slider and C is the distance between the translation axis of the slider and the rotation center of the crank. The formulation of the mathematical model for the journalbearing consists basically in using the knowledge acquired from classical fluid mechanics and its application to the lubrication of bearings with alternate motion (Hamrock et al., 2004; Gresham, 2004). Let us consider the lubricating oil completely filling the gap between the slider and connecting rod, and that the oil stock within the gap remains constant. This can be achieved by an adequate supply system whereby the oil is constantly pumped into the bearing, which works emerged in oil. Consider the journal-bearing sketched in Fig. 2. For small eccentricity, the thickness h of the oil film can be calculated as: h() = Cr (1 + ε cos )
Fig. 1. Slider, connecting rod and crank mechanism.
Fig. 2. Schematic view of the journal-bearing system and its local coordinates.
(1)
where Cr is the radial clearance, ε = e/Cr the eccentricity ratio and e is the eccentricity. In this figure Ob , Rb and Oj , Rj are the center and radius of bearing and shaft, respectively. The center line passing through Ob and Oj defines the origin of the -coordinate, with h = hmax at = 0 and h = hmin at = . Also, a local rectangular coordinate system (x, y) can be defined such that the x-axis is tangential to the bearing and the y-axis passes through the center of the bearing. 2.2. Journal-bearing kinematics Fig. 3 illustrates the kinematics of the journal-bearing system in the slider-crank mechanism. As the crank performs a complete xy (t) = −ωU0 a
As a first step in the hydrodynamic analysis an expression for xy (t) of the (x, y) coordinate system shown in the acceleration a Fig. 2 is required. This acceleration is the vector sum of the slider ˛ (t) of the cen p (t) and the tangential acceleration a acceleration a ter line. The slider acceleration can be derived from a conventional slider-crank kinematics (Doughty, 1988). In the most of practical application of a slider-crank mechanism, the ratio between the length of the crank and the conrod (R/L) is 0.25, which gives a conrod angle (˛(t)) boundary from −15◦ to +15◦ . This assumption leads to a minimum of cos ˛(t) close to 0.97 and it can be rounded to the unity for the sake of simplification of the kinematics model, allowing the focus on the lubrication model itself. In this case, the acceleration of the (x, y) system is:
L cos(ωt) ˆj − ωU0 sin(ωt) ˆ Rb
axy,r (, t) = −ωU0
L L cos(ωt) cos(˛ + ) ∼ cos(ωt) cos = −ωU0 Rb Rb
axy, (, t) = −ωU0
L L cos(ωt) sin(˛ + ) − ωU0 sin(ωt) ∼ cos(ωt) sin − ωU0 sin(ωt) = −ωU0 Rb Rb
(2)
2.3. Hydrodynamic lubrication problem turn with period T = 2/ω, the slider translates alternately along the Y-axis. At the same time, the angle ˛(t) between the vertical axis and the center line shown in Fig. 2 oscillates around the Y-axis. In the slider-crank mechanism, the constant rotation of the crank causes a translation acceleration to the slider. Thus, either the upper or the lower parts of the bearing surface can be closer to the journal’s, depending on which surface is performing work. The hydrodynamic problem is essentially the same whatever the situation considered.
The study of fluid mechanics with its physical properties and considerations leads to the proper understanding of the phenomena involved in lubrication. The present analysis is aimed at obtaining the instantaneous velocity and pressure distributions in the lubricating fluid, from which the hydrodynamic forces on the journal can be determined. Our approach is based on the analogy of the relative rotation between two cylindrical eccentric surfaces with the relative translation of two flat inclined surfaces. Stokes’
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- shaft wall:
v (h, , t) = 0 (a);
vr (h, , t) = 0 (b)
(5)
- bearing wall:
v (0, , t) = U0 cos(ω t) (a);
vr (0, , t) = 0 (b)
(6)
The solution of Eqs. (3) and (4) subject to Eqs. (5) and (6) is searched in the form:
vr (y, , t) = Vr (y, ) eiωt (a);
v (y, , t) = V (y, ) eiωt (b);
p(, t) = P() eiωt (c)
(7)
where Vr , V and P are complex amplitudes. Replacing Eq. (7) and their derivatives into Eqs. (3) and (4): −
Fig. 3. Journal-bearing kinematics.
second problem treatment (White, 1991) was then employed for the formulation in the complex domain. Consider the journal-bearing described in Fig. 2 where the journal shaft is rotating alternately in a very small gap filled with a lubricating fluid (usually oil). The basic assumptions originally proposed by Reynolds in his lubrication theory remain valid when the shaft performs an alternate motion. Accordingly, we assume that the gap is thin in comparison to the circumferential length of the bearing, h 2 Rb . As a consequence, the radial velocity component is small in comparison with the tangential component, i.e. Vr V . We also adopt the assumption of negligible inertia (Stokes flow), so that (V Cr2 )/( Rb ) 1. The fluid can be considered incompressible and possible cavitation effects are neglected. However, the flow is periodic with at time scale related to the crank period (T = 2/ω), thus the steady state flow assumption can not be adopted anymore. At this regard the goal of the present analysis is to investigate how the lubricating film flow is affected by the crank angular velocity. The assumptions of ∂/∂z = 0 and Vz = 0 is taken into account as in an infinite bearing. Thus, the flow is two-dimensional and can be represented in polar coordinates. According to Fig. 2, the radial coordinate r can be related to the y-axis through y = Rb − r, thus y = 0 at the bearing wall and y = h() at the journal wall. The angular coordinate is related to the x-axis through x = Rb since the gap is very thin. The (x, y) system is the usual coordinate frame for translational bearings. The problem consists in determining the tangential and radial velocity components defined by v (y, , t) and vr (y, , t), respectively, as well as the pressure field p(y, , t) in the lubricating fluid. From the above mentioned assumptions, the mass conservation equation can be written as: −
r-direction :
0=
V (h, ) = 0 (a);
Vr (h, ) = 0 (b);
1 ∂p ∂y
and
-direction :
∂v 1 ∂p ∂2 v + axy, (, t) = − + 2 Rb ∂ ∂t ∂y
V (0, ) = U0 (c); (9)
where Pt = P() + ωRb U0
L Rb
cos + i
(10)
The solution of Eq. (8) subject to Eqs. (9a)–(9d) is:
V (y, ) =
iω
U0 sinh (h() − y)
sinh h()
dPt iω Rb d
+
iω
⎧ ⎫ ⎨ sinh (h() − y) iω + sinh y iω ⎬ × −1 (11) ⎩ ⎭ sinh h() iω
Vr (y, ) =
U0 dh Rb d
iω
cosh y
−1
sinh2 h()
+
iω
iω Rb2
∂ ∂
dP
t
d
B(y, ) (12)
where
iω
− cosh (h() − y)
iω
+ cosh h()
−1
iω
+ cosh y
iω
−y
iω
sinh h()
(13)
The remaining condition, Eq. (9b), can be applied to Eq. (12) to give Eq. (14).
⎧ ⎡ ⎤⎫ ⎬ 3/2 d ⎨ dP 2 cosh h() iω −1 iω t ⎣ ⎦ h() − iω Rb d ⎩ Rb d ⎭ sinh h() iω
(4)
where and are the fluid’s specific mass and kinematics viscosity, respectively and axy, (, t) is given by Eq. (2). The boundary conditions are:
(8)
Vr (0, ) = 0 (d)
(3)
The Navier–Stokes equations greatly simplify thanks to the assumption of Stokes flow in the thin gap. However, a new term must be included to account for the non-inertial character of the (x, y) local coordinate system which moves with the central line. The momentum equations with negligible gravity effects become:
(b)
subject to:
B(y, ) =
∂vr 1 ∂v =0 + Rb ∂ ∂y
∂2 V 1 dPt iω − V = Rb d ∂ y2
1 ∂V ∂Vr + = 0 (a); Rb ∂ ∂y
=
iω
U0 Rdhd cosh h() b
sinh2 h()
iω
−1
(14)
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Further integration of Eq. (14) with the P(0) = P(2) = Po gives the pressure distribution as: P() = Po + ωRb U0
+ Rb U0
iω
L Rb
0
(1 − cos ) − i
tanh
h()
h() 2 iω
iω
condition
− K1 C2r
− 2 tanh
iω
d h( ) 2
(15)
iω
where
⎧ ⎫ h() iω ⎬ tanh 2 ⎨ 2 − 1 d 0 iω ⎩ h() iω −2tanh h() ⎭ 2 K1 = 2 Cr iω d 0 2 h() h()
iω −2 tanh
2
d Rb d
h3 ()
dPt,0 Rb d
= 6 U0
dh Rb d
Table 1 Some typical values for a journal-bearing system. Parameter and units
Symbol
Value
Bearing radius (m) Journal shaft radius (m) Bearing width (m) Radial clearance (gap) (m) Crank angular velocity (rad/s) Absolute viscosity (Pa s) Mass density (kg/m3 ) Crank length (m) Connecting rod length (m)
Rb Rj w Cr ω R L
10 × 10−3 10 × 10−3 15 × 10−3 20 × 10−6 <600 0.0117 887.8 0.040 0.00; 0.10; 0.20
force on the journal shaft (y = h) will be:
(16)
2
Fx (t) = wRj 0
iω
At the limiting case when ω → 0 the several imaginary parts vanish and Eqs. (14)–(16) and Eq. (14) reduces to:
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Fy (t) = wRj 0
2
∂v −psin + cos ∂y
∂v −p cos + sin ∂y
(17)
It can be seen that, for dx = Rb d, Eq. (17) corresponds to the classical equation of the Reynolds lubrication theory. These results will be used for comparison. The solution for the velocity and pressure amplitudes in the alternate bearing, Eqs. (11)–(16), is thus dependent upon three dimensionless parameters, namely: C = Cr (ω/), L/Rb and . The square of the first parameter represents the Reynolds number of the alternate motion; for C 1 the velocity profiles approach those of classical lubrication theory and the alternate bearing behaves as a conventional bearing. The L/Rb ratio is associated with the slider acceleration and exerts a further influence on the pressure distribution around the shaft. The eccentricity ratio ε can assume any value (between 0 and 1) required to keep the shaft completely lubricated. The final periodic solutions for the tangential velocity component and pressure are given in Eq. (7), whereby it can be seen that both the real and imaginary parts of the amplitudes V (y, ) and P() play an important role for determining the real part of v (y, , t) and p(, t). For small eccentricity the local instantaneous hydrodynamic
d = Fx ei(ωt+ x ) y=h
(18)
d = Fy ei(ωt+ y ) y=h
(19)
where x and y are phase angles and the directions (x, y) are indicated in Fig. 2. 3. Numerical results In the following simulations we assume given values for the three independent dimensionless parameters C , L/Rb and ε. The lubricating conditions such as pressure distribution, velocity profile and hydrodynamic force are calculated accordingly. The influence of critical parameters such as crank rotation, slider acceleration, and eccentricity are discussed. The range of the above dimensionless parameters can be determined from the reference values listed in Table 1. The simulations are strictly valid for a suitable choice of the velocity U0 satisfying (U0 Cr2 )/(Rb ) 1. Fig. 4 illustrates the amplitude and phase of the dimensionless pressure P∗ = (P − Po )/(( U0 Rb )/Cr2 ) as a function of for some typical values of parameter C at L/Rb = 0 and ε = 0.5. As can be noted in Fig. 4(a), in the 0 < C < 0.25 range the journalbearing behavior tends to that described by the classical Reynolds
Fig. 4. Dimensionless pressure around the shaft (L/Rb = 0 and ε = 0.5): (a) amplitude and (b) phase.
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Fig. 5. Dimensionless pressure around the shaft ( C = 0.25 and L/Rb = 0) (a) amplitude and (b) phase.
equations. The imaginary component depends on C and its influence in the pressure distribution is smaller in comparison with the real component. However, Fig. 4b shows that the phase inversion of 90◦ causes a very small pressure distribution in the upper lobe of the cylindrical bearing. The effects of the eccentricity ratio on pressure distribution amplitude and phase are illustrated in Fig. 5(a) and (b), respectively, for ε = 0, 0.25, 0.5 and 0.75 at L/Rb = 0 (this assumption is used only to vanish the local acceleration in the theoretical analysis) and C = 0.25. As can be observed the position of the pressure peak approaches * = as the eccentricity ratio increases. When ε = 1 the oil film is broken, causing a direct contact of journal and bearing at = . Fig. 6 shows the angular position * of the pressure peak as a function of the eccentricity for several L/Rb and C = 0.25. For L/Rb = 0, * is nearly /2 for ε = 0; the limit condition * → as ε → 1 is noted. The slider acceleration can be observed to have a significant influence at low eccentricity. Fig. 7 shows the amplitude and phase of the dimensionless velocity profiles (V (Y, ))/U0 for several angular positions around the journal shaft, where Y = y/h. Both amplitude and phase components are important in view of Eq. (7). For the selected value
Fig. 6. Angular position of the pressure peak for several L/Rb and C = 0.25.
Fig. 7. Dimensionless velocity for several eccentricities and C = 0.25: (a) amplitude and (b) phase.
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Fig. 8. Dimensionless force Fx∗ for L/Rb = 0, 10 and 20 at C = 0.25: (a) amplitude and (b) phase.
Fig. 9. Dimensionless force Fy∗ for L/Rb = 0, 10 and 20 at C = 0.25: (a) amplitude and (b) phase.
C = 0.25, the velocity amplitudes become nearly identical to the classical lubrication theory in the bearing coordinate /4 < < . However, the velocity amplitudes have negatives values from = 0 (0.6 < Y < 1.0) to = /4 (0.65 < Y < 1.0), respectively. This effect is probably caused by the alternating motion of the bearing and it depends on the rotational velocity of the crank (ω). Figs. 8 and 9 show the amplitude and phase of the force component Fx ∗ = (Fx )/((wRj U0 Rb )/(Cr2 )) and Fy ∗ = (Fy )/((wRj U0 Rb )/(Cr2 )) on the journal shaft as functions of the eccentricity for L/Rb = 0, 10 and 20 at C = 0.25. The effect of slider acceleration appears on the forces amplitudes Fx * and Fy * as seen in Figs. 8(a) and 9(a), where the effect of shear has been omitted. In this case, the increasing of L/Rb increases the forces amplitudes due to the slider acceleration, which increases the pressure peak. Notice that the force amplitude Fx * is negative and Fy * is positive (Figs. 8(b) and 9(b), respectively). To verify the time-dependence of the hydrodynamic forces behavior, some numerical results were obtained for one lap of the crank (q = ωt, where ω is constant). In this simulation, the force actuating on the bearing is proportional to the slider acceleration (kinematics condition). Figs. 10 and 11 show the amplitude and phase of the instantaneous hydrodynamic force component Fx * and Fy * for one crank lap,
considering L/Rb = 20 and C = 0.095 (ω = 3000 rpm). Furthermore, the initial angle to start the motion of the crank is 90◦ . As can be seen in Figs. 10(a) and 11(a), the hydrodynamic forces Fx * and Fy * present quite similar behavior. The hydrodynamic forces describe a continuous curve, with a parabolic tendency, from 90◦ to 270◦ and, analogously, from 270◦ to 450◦ (90◦ ). However, the behavior is different due to the inversion of the slider acceleration direction (the loading, in this case). This inversion is the cause of the discontinuity presented at approximately 270◦ in all graphs (Figs. 10 and 11). Figs. 12 and 13 show the behavior of the eccentricity ratio for one crank lap. According to Fig. 12, the eccentricity ratio reduces when the crank angle goes from 90◦ to 180◦ because the increasing of the conrod angular velocity gradient is greater than the loading gradient. Sequentially, the eccentricity ratio increases when the crank angle approaches 270◦ , because the angular velocity of the conrod vanishes. At 270◦ , the slider acceleration inversion occurs and the shaft (conrod) finds a new equilibrium position at the opposite quadrant of the bearing. Consequently, there is a discontinuity in the simulation results due to the fact that only the kinematics analysis is carried out at this point of the analysis (without inertial effects).
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Fig. 10. Instantaneous dimensionless force Fx∗ for one crank lap, L/Rb = 20 and C = 0.095 (a) amplitude and (b) phase.
Fig. 11. Instantaneous dimensionless force Fy∗ for one crank lap, L/Rb = 20 and C = 0.095 (a) amplitude and (b) phase.
Fig. 12. Eccentricity ratio behavior for one crank lap. Fig. 13. Orbit of the shaft for one crank lap.
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Fig. 13 shows the orbit of the shaft (conrod end) in the bearing for one lap of the crank. Initially, the shaft is located in the fourth (right lower) quadrant of the bearing. However, after the inversion of the slider acceleration, the shaft goes to the second (upper left) quadrant of the bearing.
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Acknowledgements The authors thank to Fapesp, CNPq and CAPES for supporting different parts of this research. References
4. Conclusions This work was aimed at the hydrodynamic analysis of lubrication for bearings with alternating rotational motion shafts, as those present in slider-crank mechanisms. The approach was based on an analogy between the relative rotation between two cylindrical eccentric surfaces and the relative translation of two flat inclined surfaces and lubrication theory assumptions were adopted. A simplified kinematics of the slider-crank mechanism was included. The problem was formulated as to determine the force on the conrod end inside the slider hole for a given eccentricity, geometry, crank rotation, slider acceleration and fluid properties. An analytical solution was obtained, which revealed three dimensionless parameters governing the hydrodynamic force, namely C , L/Rb and ε. The first parameter represents the effect of the Reynolds number related to the crank rotation, which seems to be relevant for C > 1. This situation is outside the range focused in this work, but might be important for other journal-bearing sizes and rotations. The second effect represents the (linearized) slider acceleration effect, which was shown to be important in the present numerical simulations, especially because it generates a force in the direction of the slider motion. The effect of eccentricity was observed to be strong, confirming the results of previous works. However, in a real problem, the designer wants to ensure that the shaft will be properly lubricated (i.e. ε < 1). Thus, investigations are suggested to explore the dynamics of the mechanism under loading and its connection with the conrod end position inside the slider hole, in order to reach the lubrication map of this kinetic pair.
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