Finite Element Mass-Conserving Cavitation Algorithm in Pure Squeeze Motion. Validation / Application to a Connecting-Rod Small End Bearing

Transcription

1 Finite Element Mass-Conserving Cavitation Algorithm in Pure Squeeze Motion. Validation / Application to a Connecting-Rod Small End Bearing Virgil Optasanu, Dominique Bonneau Laboratoire de Mécanique des Solides, University of Poitiers, CNRS UMR 6610, 4 Av. de Varsovie Angoulême Cedex, France Abstract A simple analytical approach using conditions of conservation of the mass is proposed for the 1-D "negative squeeze" lubrication problem in order to calculate the cavitation boundary position during oscillatory motion of two plates. The same geometrical case is analyzed using Bonneau s finite element code. Good agreements between analytical and numerical results support validation of Bonneau s algorithm. As an example of application of this algorithm to squeeze motion case, the EHD lubrication of an elastic connecting-rod small end bearing is analyzed. Influences of the shaft elasticity and lubricant piezoviscosity are presented.

2 1. Introduction Squeeze film is present in the dynamics of fluid-film bearings where loads, geometrical configuration and kinematics have important variations. Two different cases are present: "positive squeeze" and "negative squeeze" film. On the first one, "positive squeeze" appears in the case of normal approach of solid elements separated by a fluid-film and provides damping by increasing the film pressure. Thus, the "positive squeeze" is used in squeeze film dampers to reduce noise and vibrations of rotors. Its applicability in rotor dynamics and in tool-machine is considerably increasing. On the other side, the "negative squeeze" film behavior is not as well understood. Normal separation of two surfaces fully submerged in a liquid produces sudden and rapid decrease of the film pressure, which induces fluid cavitation onset and development, and after a certain period, the film collapses. In a cavitated area the fluid is a mixture of liquid, dissolved gas and liquid vapors which is temporally and spatially varying. Hays and Feiten (1964) observed cavitation formation and collapse for circular plates under non-reversing surface separation. The experimental investigation shows that cavitation expansion and collapse to a full-film can occur during surface separation. The refill of the liquid in the cavitated region is due to the pressure gradient created between ambient pressure and cavitation pressure. Rodrigues (1970), Parkins and Stanley (1982), Parkins and May- Miller (1984) and Parkins and Woollam (1986) investigated oscillatory parallel plates of varying frequency and identified classes of cavitation pattern formation which may collapses before the onset of normal surface approach. In order to solve an EHD journal bearings problem, analytical and numerical studies have been made. Fantino and Frêne (1979, 1981, and 1985) used substitution methods and plane elasticity relations for the bearing housing. To ensure convergence, under-relaxation techniques are used to limit the predicted changes in the oil film thickness at each iterative step. Oh and Goenka, (1985) use the Newton-Raphson method in conjunction with the finite element space discretisation and Murty's algorithm (Murty, 1974) to predict the non-active and full-film regions. Elrod and Adams (1975), Jones (1983), Brewe (1986), Paranjpe and Goenka (1990) and Vijayaraghavan and Keith (1990) employed mass conservative cavitation algorithms combined with finite element and finite difference algorithms to track the spatial and temporal evolution of the mixture density. The computation of pressure and density distribution is V. OPTASANU, D. BONNEAU 2

3 performed without a fixed selection of the cavitation boundary, but by local interpretation of nodal pressure and density mixture in the film. Kumar and Booker (1991a, 1991b) studied the film behavior using an Elrod like finite element formulation by considering the film as continuous with varying density and viscosity. Boedo and Booker (1995) used Kumar-Booker algorithm in the case of separating plates to study the negative squeeze action. An important point that has to be considered carefully is the location of the film reformation boundary for dynamically loaded bearings. The Reynolds boundary conditions represent acceptable conditions for the film rupture, but cannot properly satisfy the film reformation. Jacobson and Floberg (1957) and Olsson (1974) formulated boundary conditions for a moving boundary that conserve mass within the cavitated region as well as at the boundary (JFO theory). Bonneau et al. (1995) used a mass-conserving finite element algorithm to simulate the EHD bearing behavior. Mass conservation is ensured by the prediction of the cavitation region using Murty's modified algorithm associated with JFO moving condition. Validation was made for a limited set of dimensional studies and by comparison to previous finite element and finite difference results. The present study validates the finite element algorithm proposed by Bonneau et al. (1995). An analytic model is developed for the simple case of oscillating plates and the results are compared with the finite element predictions. A simple geometry is considered, i.e. two plates infinitely large with a lubricant film layer of a parabolic shape profile. The problem is solved analytically using Reynolds's boundary conditions for film rupture, flow conservation conditions for transient cavitating films and full-film reformation condition to ensure mass conservation. The influence of the dimensionless parameters on the cavitation development and collapse are analyzed. Approximate design of connecting-rod small end bearing is generally sufficient to ensure acceptable lubrication parameters. For this reason the small end bearing is less studied than the big end. The hard running conditions of new motors involve the lubrication conditions of the small end bearing more severe, which may lead to increased risks of wear and damages. Tannneau et al. (1985) studies the influence of roughness on the lubrication of such of bearings. As an application of Bonneau s numerical algorithm the case of the connecting-rod small end bearing is considered. A 3 D discretization is used for the connecting-rod and a 2 D discretization for the fluid film. The lubricant is taken as piezoviscous and the connecting-rod V. OPTASANU, D. BONNEAU 3

4 is considered elastic. The influence of shaft elasticity is also studied in order to evaluate its influence on the shape of pressure and film thickness fields. V. OPTASANU, D. BONNEAU 4

5 2. Problem formulation Figure 1 shows two rigid plates, infinitely large, separated by a thin lubricant film. These plates are considered fully immersed within a lubrication bath. The lower plate is fixed while the upper one has a parabolic shape. The oscillatory film thickness is given by : (, ) cos( ) h x t = h h ω t + a x, (1) where h 0 is the clearance at the middle point ( x = 0 ), h 1 is the amplitude of the motion, ω is the pulsation of the oscillation, and a is related to the curvature of the upper plate. Consequently, the velocity of the upper surface along z-direction is: ( ) = 1 ω ( ω ) v t h sin t (2) During oscillations, two situations are possible: a) No cavitation occurs during oscillations. b) Cavitation appears at the plate s center, grows and disappears before the end of the cycle. A third case can occur if the plates are not submerged. At the edge of the plates the film rupture is produced by ingestion of atmospheric air in the gap. This kind of cavitation has strong influence on the load capacity of squeeze dampers (Zeidan and Vance, 1989). As the plates are considered submerged in oil, only the second situation presents an interest from the point of view of cavitation, and will be studied. Figure 2 shows a top-view of the plates with the film regions. Ω is the full-film region, Ω 0 is the cavitated region, and Ω * is the cavitation boundary. When Ω * moves towards the full-film region Ω, a rupture boundary noted by Ω - develops. While when Ω * is moving towards the cavitation zone Ω 0, a reformation boundary noted by Ω + follows Mixture model and film regions. In the full film region, liquid is viewed as a monophasic incompressible medium. In the cavitation region the fluid is considered as a mixture liquid - gas/vapor. The pressure in the cavitated area is constant and equal to p cav. As the plates undergo only squeeze motion, the Couette flow is absent and the constant pressure implies that the mixture is immobile. The mixture liquid rate θ is equal to 1 in the full-film region, and it is less than 1 and associated to the pressure p cav in the cavitated region. The liquid rate in the cavitated area is time varying as follows: V. OPTASANU, D. BONNEAU 5

6 ( x, t0 ) ( x, h θ ( x, =, (3) h where t 0 is the time when the cavitation occurs at the x position i.e. the time when the pressure becomes equal to p cav at the x location. During the cavitation expansion, a finite quantity of liquid is deposited on the plates and it does not vary until the reformation occurs. When the refilling process starts, the liquid from the cavitated area participates to the reformation of the full-film. The film regions can be defined as in Table 1. Table 1. Film regions. 1., full film region 2.,, cavitation boundary 3., cavitation region The second region occupies a singular place. Four simple mechanisms for cavitation growth are considered as seen in Fig. 3. At the cavitation inception, bubbles formed by local evaporation of dissolved gases and / or liquid vapor start to grow (Fig. 3 a). When bubbles develop they start to fusion together because of the tensile stress and irregular streaks appear (Fig. 3 b). If the plates continue to separate, streaks could be broken and drops can replace them (Fig. 3 c). These drops start to deposit on the surface and a thin film (Pinkus, 1961) can be formed (Fig. 3 d). These four phenomena can subsist simultaneously in different regions of the cavitation zone. The cavitation boundary can be defined as the limit of the region where the pressure is equal to p cav, the cavitation pressure. During the reformation, the cavitation boundary separate the regions of full film and mixture liquid / gas. In the region of full-film, the pressure is higher than p cav, while in the mixture region the pressure is equal to p cav Governing equations. Due to the homogeneity of the medium upon y-direction, and neglecting the surface tension, the cavitation boundary is considered as rectilinear. Thus, a one-dimensional formulation can be applied. The governing equations are the followings: The Reynolds equation governs the full film zone. For squeeze without translating motion, it becomes: V. OPTASANU, D. BONNEAU 6

7 3 h ( ) h E p = p 12 = 0 µ t For the case of constant viscosity, Eq. (4) becomes : (4) (5) We introduce dimensionless parameters: H = h h 0 P = p p amb x X = T x a and Reynolds equation becomes: t = ω and 2π The Boundary conditions are the followings: p At the ambient boundary x 2 xa 2 0 pamb µ ω µ = (6) 2π h = x a ( x a is constan = p amb ( P = 1) (8) At the cavitation boundary x = x c ( x c is time-dependen p =. ( P = P ) (9) cav p cav At the cavitation boundary x = x c, only during cavitation growth we have moreover Reynolds's condition p x = 0 ( P = 0 ). (10) X The mass conservation condition is given by the flow balance throughout the reformation boundary. The flow rate in the active zone near the boundary is given in the squeeze case by the Poiseuille velocity profile across the film thickness : 3 h ( ) = p Q Ω ± 12µ For a rupture boundary, the flow is zero, as p = 0 (the Reynolds condition at this interface). In the case of a reformation boundary, the balance of the flow on both sides of the interface full film / cavitation zone gives the velocity of the reformation boundary :. Q( Ω + ) x c = (12) h (1 θ ) (7) (11) V. OPTASANU, D. BONNEAU 7

8 3. Analytic algorithm According to the hypothesis, three periods are considered during oscillations: (i) Cavitation not yet appearing. The pressure is calculated using Eq. (A1) presented in the Appendix. When, during the plates separation, the calculated pressure at the plate center becomes less or equal than p cav, then cavitation occurs; (ii) Cavitation area is evolving. During the expansion of the cavitation region the quantity of liquid found at the cavitation boundary location is deposed on the solid surfaces and later this liquid will participate to the reformation. Numerically, the film thickness at the cavitation boundary is memorized, in order to determinate the liquid rate later. When the cavitation region stops its growth the reformation starts; (iii) Cavitation area is reducing. The full-film is reforming and the quantity of liquid left on the surfaces in (ii) participates to this reformation. When cavitation region vanishes the full-film occupies the whole domain and the process restarts. The analytic pattern leads to a differential equation, the variable of which is the position of the cavitation boundary. The Appendix contains the analytic development. The final form of this equation is:. (, ) x = F x t (13) c i c where the functions F i (i=1 for cavitation growth and i=2 for cavitation diminution) are detailed in Appendix. Equation (13) is solved using a fourth-order Runge-Kutta integration method with variable time step and error control. The influence of different parameter variations on the dimensionless cavitation interface location is studied. In view of simplicity, the curvature of the plates was set to zero ( a = 0 ); consequently, parallel plates have been analyzed. The cavitation pressure was considered as zero. Two dimensionless parameters affect the dimensionless cavitation boundary position. These are the dimensionless viscosity ( µ ) and the relative amplitude of the movement h 1. h0 Keeping constant the ratio h h 1 0 and modifying ( µ ), the extent of the cavitation region and the moment of the cavitation collapse are changing. Parameters modification generate a V. OPTASANU, D. BONNEAU 8

9 variation of dimensionless viscosity µ, as seen in Eq. (6). Increasing µ (by increasing µ, ω, or x a ), the extent of the cavitation region and the time of cavitation presence increase too, as seen in Fig. 4. The physical interpretation of these variations can be carried out as follows : an increase of µ is equivalent (from point of view of the dimensionless cavitation interface position) with a decrease of p amb (other parameters being kept constan, which means a lower pressure gradient influence. As a result, the maximum area covered by the cavitation becomes more important. During reformation, a lower pressure gradient leads to a lower reformation boundary velocity, as seen in Eq. (12), i.e. the refill process is slower. If µ is kept unchanged and the amplitude of the motion is increased, the area covered by the cavitated zone becomes larger because of the growth of the normal separation velocity and the diminution of the minimum film thickness (Fig. 4). Boedo and Booker (1995) studied the case of separation of circular and square plates at constant velocity. Using dimensions representative to those used by Boedo and Booker (1995), the results shown in Fig. 5 have been obtained. The cavitation inception commences very shortly after the onset of plates separation, and the total process of growth and collapse have a short duration. Cavitation vanishes at T 0.1. Similar results are obtained by Boedo and Booker (1995) for circular plates. Further, the analytic algorithm is used for comparison with results obtained using the finite element algorithm in order to validate the Murty's modified algorithm (Bonneau et al., 1995). 4. Finite element algorithm The finite element algorithm has been previously described (Bonneau et al., 1995). The present squeeze problem is treated using the 1-D version of the code. For the prediction of the cavitation region a modified Murty's algorithm is used Modified Murty s algorithm. The problem of determination of active film region and cavitation film region is treated as a complementarity problem by the Murty s algorithm (1974). As the latter does not ensure the conservation of the mass, a supplementary condition is introduced to determine the boundaries location. V. OPTASANU, D. BONNEAU 9

10 Bonneau et al. (1995) modified the Murty s algorithm by adding flow conservation conditions throughout the reformation and rupture full film boundaries. Let Ω and Ω 0 be, respectively, the active and the cavitation region, Ω, Ω respectively the reformation and the + rupture boundaries. The modified Murty s algorithm can be summarized as follows : Identification of a cavitated zone Ω 0, with p=0 for all nodes in Ω 0. Solution of Reynolds equation in the active region Ω. Each node with p<0 will be placed in Ω 0 and the corresponding pressure will be set to zero. Verification of E(p)<0 in Ω 0. Nodes not verifying this inequality are placed in Ω. The process is repeated until a stable cavitation zone is obtained. A supplementary condition is introduced by Eq. (13). This velocity allows the prediction of a new position of the reformation boundary from the position at the previous time step: (14) with t n the n-th time step and the position of the reformation boundary at time t n. This represents a restrictive condition for the placement of the nodes on the full film. Only nodes placed in the interval x c, x c can be placed in Ω. This represents a restrictive t n t n 1 condition which, combined with the Murty s algorithm, leads to a mass conserving algorithm for prediction of the cavitation regions Finite element algorithm results. The cavitation problem for oscillating parabolic plates is treated using the analytic and finite element models, with the same geometrical parameters. Fig. 6 shows results for the plates system whose geometrical description is presented in Table 2. Agreement between the analytic and finite element results is good. Differences are due to the finite element mesh; in this case the cavitation interface can take only discrete values. As seen, in both cases the cavitation collapses well before the onset of plates approach, during the surfaces separation, as also observed by Parkins and Woollam (1986). Table 2. Geometric characteristics and rheologic properties of the lubricant h µm h µm ω rad.s -1 x a mm µ 5 mpa.s V. OPTASANU, D. BONNEAU 10

11 p amb 10 6 Pa p cav 0 Pa a 10-7 m -1 The mixture liquid rate history at the plates center is shown in Fig. 7 and compared to the analytic results. The liquid rate has a continuous decrease until the reformation moment. When reformation occurs, the mixture liquid rate value has an abrupt rise to 1. Figure 8 represents the pressure time evolution. After the onset of plates approach (near T=0), the central pressure falls to zero and the cavitation zone starts to grow. First, it grows rapidly; then, it slows down its expansion, and it reaches its maximum extent well before the onset of plates approach. While the plates separate, the pressure value remains under the ambient pressure. When the plates start to approach, the pressure raises. When plates continue to approach, the pressure increases rapidly. At the end of the motion, when the approach velocity of the surfaces vanishes the pressure becomes equal to the atmospheric pressure and a new cycle is restarted. 5. Application to an EHD study of a connecting-rod small end bearing The connecting-rod small end bearing needs particular hypothesis in order to modelize the phenomena within the fluid film as realistically as possible. In a connecting-rod small end bearing case, as the sliding velocity is very low, sliding hydrodynamic effects are negligible. Thus, the relative motion of the shaft within the bearing is considered as a squeeze motion, without rotation. It is generally accepted that the shaft rotates only within the piston holes. Computations were also made considering an alternative motion of the shaft. Results were quite the same as for pure squeeze motion. The film thickness is given by : h = h 0 + (15) h el where h 0 is the nominal film thickness due to the eccentricity and h el is the sum of the elastic deformations of the bearing and of the shaft, which are due to the lubricant pressure. The elastic deformations are obtained using two compliance matrices calculated by FEM method. The bearing and shaft are meshed with 20 nodes isoparametric elements. An 8 nodes isoparametric element mesh of the inner bearing surface is constituted of the faces of the 3-D elements located on the housing. This mesh is also used to discretize the Reynolds equation V. OPTASANU, D. BONNEAU 11

12 using the finite element method (Bonneau et al., 1995). The elastic deformation at node i is given by : h i = n k = 1 b k n C ( i, k) f + C ( i, k) f (16) k= 1 sh k where C b (i,k) and C sh (i,k) are the radial displacement at node i given by a unit load at node k applied on the bearing and on the shaft, respectively. Considering the meshes presented Fig. 9, both connecting-rod and shaft present a symmetry following X-Y plane, which means no displacement along Z axis of nodes located in this plane. More, the nodes at the end of the connecting-rod are completely blocked. When assembled, the shaft is longer than the housing, and only displacements along Z axis are allowed for the nodes of the shaft surface located inside the piston housing (not represented). Assuming that inertia forces are negligible in the film, the balance of the applied loads with the hydrodynamic pressure is : S p = cos θ dθ F ; x p θ dθ = Fy S sin (17) The period of applied load F on the connecting rod small end bearing is 720 degrees. F represents reciprocating and combustion forces. Due to the important pressures developed in the film, a piezoviscous behavior of the lubricant must be considered. The variation of the viscosity with respect to the pressure is assumed to be (Prat et al., 1994) : 0 ( + c p) 1 µ = µ (18) 1 c 0 2 µ is the viscosity at p = 0, c 1 and c 2 are two characteristic coefficients of the lubricant. Several simulations have been performed for the connecting-rod small end bearing. Table 3 shows the geometric characteristics and lubricant rheologic properties. The finite element mesh of the half bearing is presented Fig. 9. A 711 nodes mesh is used to discretize the Reynolds equation for the oil film. Fig. 10 shows the load diagram at 2000 rev/min. The supply holes are located at ±120 degrees from the longitudinal axis of the connecting-rod. Table 3. Bearing characteristic and rheologic properties of the lubricant Diameter of the bearing 12.5 mm Radial clearance 12.5 µm Length of the bearing (variable) 16 to 25 mm Pressure supply 0.1 MPa Lubricant viscosity at p= Pa.s Piezoviscosity coefficient c Piezoviscosity coefficient c MPa -1 V. OPTASANU, D. BONNEAU 12

13 5.2. Elastic effects. Several cases are considered. Figure 11 shows the influence of the elasticity of the bearing and shaft on the maximum pressure in the film. If both shaft and bearing are considered rigid, due to the non compliance of the surfaces, the pressure field is very localized and the calculated maximum pressures could be larger than 3000 M Pa, which is not compatible with the hypothesis of the bearing rigidity. In the case of a rigid shaft and an elastic bearing, the maximum pressure rises to only 640 MPa, which still remains significant. If both shaft and bearing are considered elastic, the calculated maximum pressure reaches values about 580 MPa, but only in a few points, near the edges of the bearing. Figure 12 shows the film pressure, the film thickness and the radial deformations of the set bearing / shaft. If both bearing and shaft are considered elastic, despite of the peaks of pressure noted on the edges of the bearing, the pressure is less important than for the case with a rigid shaft and elastic bearing. This result is explained by a more important compliance of the surfaces in both axial and circumferencial direction. In the high load regions, the calculated film thickness is globally higher if the shaft is considered elastic, which induces a less important risk of solid-to-solid contact Piezoviscous effects. Figure 13 shows the influence of the piezoviscous effect on the minimum film thickness at 2000 rev/min. One can note that for the piezoviscous case the values are globally higher than those obtained if this effect is neglected. However, for a pure squeeze motion, the increase of the viscosity with the pressure has lower effects than in a sliding motion. In pure squeeze motion, the increase of the viscosity makes slower the fall of the thickness but does not prevent it. Figure 14 shows two details of the thickness fields in piezoviscous and constant viscosity case for both elastic shaft and bearing. If the viscosity is constant, the thickness falls down on the edges of the bearing, which prevent axial flows and ensure the lift of the shaft within the bearing. For the piezoviscous case the rise of the viscosity value on the bearing edges, where the pressure is important, ensures a larger film thickness. This increase of the viscosity avoids the oil leakage, despite of a thicker oil film. Thus, the same load is supported with less risks of solid-to-solid contact. For a rigid shaft, the calculated film thickness in the hard-loaded region is globally 2 times higher with piezoviscous assumption than with isoviscous hypothesis. 6. Conclusions V. OPTASANU, D. BONNEAU 13

14 The paper presents a comparison between the results of a simple, 1-D analytic algorithm and a mass-conservative FEM algorithm (Bonneau, 1995) for the case of oscillating parabolic plates. The good agreement between results validate the finite element and the correctness of the mass-conserving cavitation model. Following the parameter values, the cavitation may vanish well before the onset of plates approach, which is in agreement with previous experimental investigations. It is shown that the general shape of the interface position and mixture density history depend only on the relative amplitude of oscillation and on the dimensionless viscosity parameter. As an application, the problem of the lubrication of a connecting-rod small end bearing is analyzed. This is a high load machine element and it represents a good example of a squeeze film application. Considerations of shaft elasticity is necessary for the connecting-rod small end bearing. Taking into account the shaft elasticity leads to completely modified pressure fields. The maximum pressure is displaced from the center of the bearing to its edges. The film thickness is sensibly modified too ; it is globally higher in the hard loaded regions. The piezoviscous effects leads to an increase of the film thickness in the high load zones, which represents better lubrication conditions. V. OPTASANU, D. BONNEAU 14

15 Nomenclature a = parabolic plate curvature [m -1 ] h = film thickness [m] p = pressure [Pa] p amb = ambient pressure [Pa] p cav = cavitation pressure [Pa] t = time [s] x a = ambient interface position [m] x c = cavitation interface position [m] H = dimensionless film thickness [-] P = dimensionless pressure [-] T = dimensionless time [-] X c = dimensionless cavitation interface position [-] µ = viscosity [Pa s] µ = dimensionless viscosity [-] θ = mixture liquid rate [-] ω = the pulsation of the oscillation [s -1 ] Ω = full-film region Ω 0 = non-active region Ω = rupture boundary Ω = reformation boundary + References Boedo, S., and Booker, J. F., 1995,"Cavitation in Normal Separation of Square and Circular Plates," ASME JOURNAL OF TRIBOLOGY, Vol. 117, pp Bonneau, D., Guines, D., Frêne, J., and Toplosky, J., 1995, "EHD Analysis, Including Structural Inertia Effects and Mass-Conserving Cavitation Model," ASME JOURNAL OF TRIBOLOGY, Vol. 117, pp Fantino, B., Frêne, J., and Duparquet, J., 1979, "Elastic Connecting-Rod Bearing with Piezoviscous Lubricant: Analysis of the Steady State Characteristics," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 101, No.2, pp Fantino, B., 1981, "Influence des défauts de forme et des déformations élastiques des surfaces en lubrification hydrodynamique sous charge statiques et dynamiques," Thèse No. 1- DE-8122, INSA de Lyon, France. V. OPTASANU, D. BONNEAU 15

16 Fantino, B., Godet, M., and Frêne, J., 1983, "Dynamic behaviour of an Elastic Connecting-rod Bearing Theoretical Study," Proceeding of the Automotive Engines, Studies of Engine Bearing and Lubrication SP-539 NO , pp Fantino, B., and Frêne, J., 1985, "Comparison of Dynamic Behaviour of Elastic Connecting-rod Bearing in Both Petrol and Diesel Engines," ASME JOURNAL OF TRIBOLOGY, Vol. 107, pp Hays, D. F., and Feiten, J. B., 1964, "Cavities Between Moving Parallel Plates," Cavitation in Real Liquids, R. Davies, ed., Elsevier Publishing Company, New York, N.Y., 1964, pp Jacobson, B., Floberg, L., 1957, "The finite Journal Bearing Considering Vaporisation," Chalmers Tekniska Hoegskolas Hnndlingar, Vol. 190, pp Kumar, A., and Booker, J. F., 1991a, "A Finite Element Cavitation Algorithm," ASME JOURNAL OF TRIBOLOGY, Vol. 113, pp Kumar, A., and Booker, J. F., 1991b, "A Finite Element Cavitation Algorithm: Application/Validation," ASME JOURNAL OF TRIBOLOGY, Vol. 113, pp LaBouff, G. A., and Booker J. F., 1985, "Dynamically Loaded Journal Bearings: A Finite Element Treatment for Rigid and Elastic Surfaces," ASME JOURNAL OF TRIBOLOGY, Vol. 107, pp Murty, K. G., 1974, "Note on a Bard-type Scheme for Solving the Complementarity Problems," Opsearch, Vol. 11, pp Oh, K. P., and Goenka, P. K., 1985, "The Elastohydrodynamic Solution of Journal Bearings under Dynamic Loading," ASME Journal of Tribology, Vol. 107, pp Olsson, K., 1974, "On Hydrodynamic Lubrication with Special Reference to Nonstationary Cavitation," Chalmers University of Technology, Goteborg. Parkins, D. W., and Stanley, W. T., 1982, "Characteristics of an Oil Squeeze Film," ASME JOURNAL OF TRIBOLOGY, Vol. 104,pp Parkins, D. W., and May-Miller, R., 1984, "Cavitation in an Oscillatory Oil Squeeze Film," ASME JOURNAL OF TRIBOLOGY, Vol. 106, pp Parkins, D. W., and Woollam, J. H., 1986, "Behavior of an Oscillating Oil Squeeze Film," ASME JOURNAL OF TRIBOLOGY, Vol. 108, pp Pinkus, O. and Sternlich, B., 1961, "Theory of Hydrodynamic Lubrication," McGraw-Hill. Prat, P., Vergne, Ph., Sicre, J., 1994, "New Results in High Pressure and Low Temperature Rheology of Liquid Lubricant for Space Application", ASME JOURNAL OF TRIBOLOGY, Vol. 116, pp Rodrigues, A.N., 1970, "An Analysis of Cavitation in a Circular Squeeze Film and Correlation with Experimental Results," Ph. D. thesis, Cornell University, Ithaca, NY. Tanneau, G., Frêne, J., Berthe, D., 1985, Theoretical Approach to Roughness Effects in the Small-End Bearing of a Connecting-Rod, Proceeding of the 11th Leeds-Lyon Symposium on Tribology, Butterworths, pp Zeidan, Y.F., Vance, J.M., 1989, Cavitation Leading to a Two Phase Fluid in a Squeeze Film Dapmer, Tribology Transaction, Vol 32, 1, pp V. OPTASANU, D. BONNEAU 16

17 z y h x 6. Fig.1. Geometry of the parabolic plates V. OPTASANU, D. BONNEAU 17

18 y Ω Ω x Ω Ω 0 Ω Fig. 2. Film regions : Ω 0 = cavitation region, Ω = full film region, Ω = cavitation boundary V. OPTASANU, D. BONNEAU 18

19 Ω Ω a) Ω Ω b) Ω Ω c) Ω Ω d) Fig. 3. Four stages for the cavitation growth. V. OPTASANU, D. BONNEAU 19

20 h h 1 = 0 h h = 0 _, µ = _, µ = 0. 5 a) b) c) Xc h h 1 = _, µ = T Fig.4. Dimensionless cavitation boundary position for several values of parameters V. OPTASANU, D. BONNEAU 20

21 0.8 X a Xc h µm h µm ω 1.9 rad/s x a 25.4 mm µ Pa.s p amb 10 5 Pa p cav 0 Pa a 0 mm -1 h T 7. Fig. 5. Dimensionless cavitation interface position V. OPTASANU, D. BONNEAU 21

22 Cavitation interface position FEM algorithm Analytic algorithm Xc T Fig. 6. Comparison between results on cavitation interface position V. OPTASANU, D. BONNEAU 22

23 FEM algorrithm Analytic algorithm θ T 8. Fig. 7. Comparison between history of mixture liquid rate at the center of the plates V. OPTASANU, D. BONNEAU 23

24 Fig. 8. Pressure time-evolution V. OPTASANU, D. BONNEAU 24

25 Z X Y 9. Fig. 9. Finite element mesh for the half connecting-rod small end and its shaft V. OPTASANU, D. BONNEAU 25

26 50 Fy (N) Fx (N) 10. Fig. 10. Polar load diagram at 2000 rev/min V. OPTASANU, D. BONNEAU 26

27 4000 Pressure (M Pa) Both rigid shaft and bearing Elastic bearing and rigid shaft Crank angle (degrees) 11. Figure 11. Maximum pressure evolution V. OPTASANU, D. BONNEAU 27

28 12. rigid shaft and elastic bearing elastic shaft and elastic bearing MPa MPa Pressur e mm Pressure mm mm mm mm m 0. 0 m Film thickness mm Film thickness mm mm mm Radial deformations mm Radial deformations mm mm mm mm mm Fig. 12. Comparison on film pressure, film thickness and radial deformations at 370 degrees crank shaft. V. OPTASANU, D. BONNEAU 28

29 Hmin (mm) 1.2E-2 8.0E-3 4.0E-3 0.0E+0 µ = µ = µo*(c2*p)^c1 µ ( + c 1 µ = µo µ = µ 0 1 c Crank angle (degrees) 17. Figure 13. Influence of the piezoviscosity on the minimum film thickness at 2000 rev/min. V. OPTASANU, D. BONNEAU 29

30 1.3 P max = 400 MPa 1.1 H min = 0.1 µm piezoviscosity P max = 443 MPa 1.2 H min = 0.1 µm isoviscosity Fig. 14. Influence of the piezoviscosity on pressure and film thickness fields for elastic assumptions on both connecting rod and shaft (at 360 degrees crank shaf V. OPTASANU, D. BONNEAU 30

31 Appendix After two integration of the Reynolds equation, Eq. (4), considering the geometrical configuration, we obtain the analytical formula of the pressure : 3µ p = ah where (, ) I x t v( ( x, 1 3 = dx = h x t with b( h( ( x, I( x, B( x + A + 2 (, ) 8 b( c c, b( a arctan x a b t 3 x + + ( ) 8 b( h( x, 4 b( h( x, = 0,. Moreover, A( x c, and ( x from Eq. (8) and Eq. (9) A and B ( x c ( x ( 3µ v 1 1, = pamb pcav + 2 a h a c v( ( x, 3 c, = µ A c c, 2 ah c ( x, I( x x (A1) B c, are defined using boundary conditions 2 ( x, h( x, I( x, I( x, For the cavitation growth, the equation of the interface position is obtained by temporal derivation of Eq. (10) and separation of variables. Thus the following equation is obtained : ( ) x ( F x x ( c a c t = 1,, (A2) where F ( x, y, g 1 1 ( x, y, = = (,, ) + 12 µ ( ) g ( x, y, + 12 µ v( g x y t acc t y 2 1 3µ v( 1 1 a h( x, h( y, (, ) I( y, p + p amb cav ( I x t ) 2 2 g ( x, a 1 12 µ v( x (, ) (, ) (, ) + h x t I x t I y t c ( ) g 2 ( x, y, = 3µ v( 1 1 a h( x, h( y, (, ) I( y, p + p amb cav ( I x t ) 2 2 µ acc( v( 2 a 2 h( x h( y µ I( x, I( y, ( ) ( g ( x, g ( y, ) , a, h x, t h y, t + ( ) ( ) 2 2 g ( x, a x 1 = + 4b t 3 2 a x ( ) h( x, 4 b( h( x, 4 b( h( x, b ( h( x, 2 3 V. OPTASANU, D. BONNEAU 31

32 b( 15 v( a a 15 v( x 5 v( x v( x g ( x, = arctan x 16 vb( b( 16 b( h( x, 8 b( h( x, 2 b( h( x, acc t = is the acceleration of the upper plate. t where ( ) For the case of cavitation collapse, the equation that gives the boundary velocity comes from Eq. (12), by replacement of p ( ) c( ) a c( ) x x = xc : x t = F 2 x, x t, t (A3) with (, ( ), ) F x x t t 2 a c (,, ) ( ) ( ( ) ) ( ) c( ) a c( ) µ h x ( θ x t, t 12µ v t x t + A x x t t = 12 1 c [ c ] V. OPTASANU, D. BONNEAU 32