Hydrodynamic and elastohydrodynamic studies of a cylindrical journal bearing

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2010,22(2):155-163 DOI: 10.1016/S1001-6058(09)60041-X

HYDRODYNAMIC AND ELASTOHYDRODYNAMIC STUDIES OF A CYLINDRICAL JOURNAL BEARING* ATTIA HILI Molka, BOUAZIZ Slim, MAATAR Mohamed, FAKHFAKH Tahar, HADDAR Mohamed Mechanics Modelling and Production Research Unit, (U2MP), National School of Engineers of Sfax (ENIS), BP.1173, 3038, sfax, Tunisia, E-mail: [email protected]

(Received August 27, 2009, Revised January 18, 2010)

Abstract: In this article, the effect of the bearing elastic deformation on the performance characteristics of a cylindrical journal bearing is analyzed. The variety of simulation models covers hydrodynamic (HD) and elastohydrodynamic (EHD) lubrication theories. The Reynolds equations governing the flow in the clearance space of the journal bearing are obtained by considering the effect of mass transfer across the fluid film. The finite element method with an iteration scheme was employed to solve both the Reynolds equation and the three-dimensional elasticity equation representing the displacement field in the bearing shell. The converged solutions for the lubricant flow and elastic deformation vector are obtained. Dynamic characteristics of the journal bearing are computed for HD and EHD theories. Numerical simulation results show that the flexibility of bearing liner has a significant influence on the performance of a cylindrical journal bearing. Indeed, the elastic deformations of the bearing liner extend the pressure area in the bearing and increase the minimum film thickness. Although, dynamic coefficient, load capacity and attitude angle decrease. Key words: Reynolds equation, hydrodynamic (HD) and elastohydrodynamic (EHD) lubrication theories, dynamic coefficients, relative eccentricity, deformation coefficient

1. Introduction  Elastic deformation of the bearing liner under hydrodynamic pressure changes the fluid film profile, modifies the pressure distribution and therefore changes the performance characteristics of journal bearings. Some investigations in the field of tribology have been directed to hydrodynamic (HD) and elastohydrodynamic (EHD) analysis because significant changes in the bearings performance characteristics have been observed due to the flexibility of bearing liner under heavy load.  * Biography: ATTIA HILI Molka (1979-), Female, Ph. D., Assistant Professor Corresponding author: BOUAZIZ Slim, E-mail: [email protected]

Therefore, to obtain an optimum design of a journal bearing system, the flexibility of bearing liner must also be considered along with the bearing geometric and operating parameters. A comparison between HD and EHD theories should be made. When bearings are subjected to heavy load, the bearing deformation may quite often affect the clearance space geometry of the bearing. Thus, performance characteristics become significantly different from those computed in HD theory where the bearing liner is assumed to be rigid. Several literatures are available on the HD and EHD analysis of cylindrical journal bearing for Newtonian lubricants[1-3]. Byoung and Kyung determined the static characteristics of a journal bearing with micropolar fluids[4]. Das et al.[5] presented the dynamic characteristics of hydrodynamic journal bearings lubricated with micropolar fluids. The modified Reynolds equation was derived using the

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micropolar lubrication theory. By applying the first-order perturbation of the film thickness and steady-state film pressure, the dynamic characteristics in terms of the components of stiffness and damping coefficients, critical mass parameter and whirl ratio were obtained. Liu et al.[6] solved simultaneously the governing Reynolds and elasticity equations by using both the finite difference and finite element methods. The predicted transient film thickness was compared with the estimation based on the quasi-static analysis. Narayanan et al.[7] determined static and dynamic characteristics of circular rigid bearings with micropolar fluids. Jerzy and Rao[8] investigated the variation of nonlinear stiffness and damping coefficients in a journal orbit with respect to its equilibrium position. Sukumaran Nair and Prabhakaran Nair used the finite element technique to solve the modified Reynolds equation. The lubricating oil containing additives and contaminants was modelled as micropolar fluid[9]. Recently, Prabhakaran Nair et al.[10] showed that for any eccentricity ratio and deformation coefficient, load-carrying capacity of a two-lobe journal bearing increases with the increase in volume of concentration of additives. Furthermore, for a fixed value of mass transfer rate the load-carrying capacity increases with increase in volume concentration of additives at any value of deformation coefficient and eccentricity ratio. The existing literature shows that most authors concerned with the study of static and dynamic performances by changing the geometry of the bearing or the characteristics of the lubricant. In the present work, performance characteristics in terms of pressure distribution, film thickness, dynamic coefficients, carrying load capacity and attitude angle of HD journal bearing are computed and compared to EHD lubrication. The effect of deformation of the bearing liner on these characteristics is further analyzed.

Fig.1 Geometry of hydrodynamic journal bearing

2. Governing equations 2.1 Reynolds equation The hydrodynamic pressure and the bearing deformation effects are interdependent, the problem involves simultaneous solution of the Reynolds equation and three-dimensional linear elasticity equations. The geometry of a hydrodynamic journal bearing is shown in Fig.1. Under the hypothesis of thin fluid film, it is assumed that the flow is laminar and that inertia is negligible. The fluid is Newtonian and incompressible, and the density, specific heat, thermal conductivity and heat transfer coefficients are assumed to be constant. The oil pressure distribution is obtained by solving numerically the generalized Reynolds equation is given by 3 wp º 3 wp º w ª w ª h + ur h + ur + O2 =   « » « wT ¬ wT ¼ wz ¬ wz ¼»

6

w  h + ur wT

+12

w  h + ur

(1)

wW

where

z=

z h R , h = , O = , W = Zt , C C L

h = 1+ H cos T , H = p §C · p= PZ ¨© R ¸¹

e , C

ur =

ur , C

2

with x and y are the coordinates in the lubricating plane, C is the radial clearance, h is the oil film thickness, R and L are respectively the bearings radius and length, Z is the angular velocity of the journal, t is the time, T is the angular coordinate, e is the bearing eccentricity, ur is displacement of bearing liner in the radial direction, P is the lubricant viscosity and p is the oil film pressure. The parameter with a bar on its top indicates a non-dimensional parameter. 2.2 Finite element formulation The flow field in the clearance space of the

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bearing is discretized into four nodded isoparametric elements[11,12]. The bearing liner is divided into 72 elements in the circumferential direction and 22 elements in the axial direction. The discretization of the flow field is given by Fig.2.

the elasticity matrix, “ e ” = 1, 2,! , ne and ne denotes the number of element. The displacement vector of the bearing liner is

­uT (r ,T , z ) ½ ^G ` = °®ur (r ,T , z ) °¾ °u ( r , T , z ) ° ¯ z ¿

(3)

where uT , ur and u z are displacements of bearing liner respectively in the circumferential, radial and axial directions. The displacement field in non-dimensional form can be written as Fig.2 Discretization of the flow field

­uT ½ e ° ° {G } = ®ur ¾ = [ N ] j ^d ` °u ° ¯ z¿ j

The bearing liner is a finite length cylinder subjected to hydrodynamic loading due to the fluid film pressure on its internal surfaces. The distribution of fluid film pressure is such that it deforms the bearing in all directions. The bearing liner is discretized by eight nodded hexahedral isoparametric elements. The discretization of the bearing liner is shown in Fig.3.

(4)

where [ N ] j is the element shape function matrix and

^d `

e

is the nodal displacement vector of an

element, given by

^d ` = > u e

, ur1 , u z1 ,....., uT n , urn , u zn @

T

T1

(5)

in which n is the number of nodes per element. The element equations are obtained by minimizing potential energy for each element. So we can write

­ G E pe ½ ° ° ° wui ° e ° ne ° °G E p ° ® ¾=0 ¦ e =1 ° wvi ° °G E e ° ° p° °¯ wwi °¿

Fig.3 Discretization of bearing liner

Using the linear elasticity theory, the expression of the potential energy of an element when bearing is subjected only to surface traction forces T , is written as[13]

Ep =

³³ [T ] {G } rdT dz e

S

(6)

1 [G ]e [ J ]Te [ D]e [ J ]e {G }dV  2 ³³³ Ve e

Using the above conditions, the equation system is reduced to (2)

j

where {G } is the displacement vector of the bearing liner, [T ] is the traction forces, [ J ] is the operator matrix relating to strains and displacements, [ D] is

ne

¦ ª¬ ³³³ [ J ]

Te

[ D]e [ J ]e rdT drdz 

e =1

³³ > N @ >T @ Te

e

rdT dz º ¼

0

(7)

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The element equation may be written in the matrix form as

[k ]e ^d ` = ^ F ` e

e

(8)

e

>F @

e

ne

W x = ¦ ³³ p e cos T dT dz

where

>K @

2.4 Load capacity After giving the pressure field, the nondimensional load capacity is obtained by using the following relations: (13)

e =1

= ³³ [ J ]Te [ D]e [ J ]e rdșdrdz = ³³ > N @

Te

>T @

e

(9)

rdT dz

(10)

[ k ] ^d ` = C d ^ F `

(11)

where [k ] is the stiffness matrix,

^d `

is the

displacement vector, Cd = PZ  R / C eb / Eb R is a non-dimensional deformation coefficient, which takes into account the flexibility of the bearing liner[10], with eb being the bearing liner thickness and Eb being 3

^F `

is the nodal force vector

generated by the pressure field. 2.3 Boundary conditions The bearing liner is assumed to be contained in a rigid housing (see Fig.1). Thus the outer surface of the bearing liner, which is in contact with the inner surface of the housing, does not deform, implying that the nodes in contact with the rigid surface are restrained from moving.

­uT i ½ ­0 ½ ° ° ° ° ® u r i ¾ = ®0 ¾ ° u ° °0 ° ¯ zi ¿ ¯ ¿

(14)

e =1

By using non-dimensional form, the following global system equation is obtained:

Young’s modulus,

ne

W y = ¦ ³³ p e sin T dT dz

(12)

where “ i ” is the number of nodes on the bearing liner of rigid housing interface. The following boundary conditions are used to obtain the pressure field in the bearing: (1) Pressure at the journal bearing edges is equal to zero: p >T , z = r( L / 2)@ = 0 . (2) By using the Reynolds hypothesis, the pressure is zero in the areas where the pressure gradient in the circumferential direction becomes zero: p T , z = 0 with wp/wT T , z = 0 . The application of these conditions imposes the use of the Christopherson assumption[14] negative pressure is set to zero in each iteration.

The load carrying capacity is given by





2 1/ 2

2

W = W x +W y

(15)

The angle between the central line and the load line is known as attitude angle and is given by

tan ) = 

Wy Wx

(16)

2.5 Fluid film stiffness coefficients The non-dimensional fluid coefficients are defined as[10]

ª Axx «A ¬ yx

ª wW x « Axy º wx = « » « wW y Ayy ¼ « ¬« wx

film

stiffness

wW x º » wy » wW y » » wy ¼»

(17)

where W x and W y are the non-dimensional film force components respectively in the x- and y-directions. 2.6 Fluid film damping coefficients The damping coefficients are defined as[10]

ª Bxx «B ¬ yx

ª wW x « Bxy º wx = « » « wW y Byy ¼ « ¬« wx

wW x º » wy » wW y » » wy ¼»

(18)

where x and y are the journal centre velocities respectively in the x- and y-directions. 3. Solution procedure 3.1 HD analysis The finite element method is used

to

solve

the

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Table 1 System parameters Parameter Oil dynamic viscosity, P Radial clearance, C Bearing radius, R Bearing length, L Angular velocity of the journal, Z Stiffness coefficients, Axx , Axy , Ayx , Ayy Damping coefficients, Bxx , Bxy , Byx , Byy Time, t Bearing eccentricity, e Relative eccentricity, H Oil film thickness, h Minimum fluid film thickness, hmin

used to compute the pressure field for a rigid bearing. Performance characteristics in terms of load capacity, minimum film thickness, stiffness coefficients, damping coefficients and attitude angle are calculated at various relative eccentricities. 3.2 EHD analysis For EHD analysis, an iterative process is repeated until the required convergence is achieved. The converged nodal pressures are then used to calculate the nodal displacements. The film thickness is modified by considering the radial component of the nodal displacements to get the solution of the nodal pressures. Iterations are also required to obtain performance characteristics for a wide range of values of the deformation coefficient which takes into account the flexibility of the bearing liner. 4. Results and discussions The static and dynamic performance characteristics of a rigid cylindrical journal bearing with Newtonian lubricants are computed and compared with the EHD theory taking into account the flexibility of the bearing liner. System parameters are presented in Table 1.

Oil film pressure, p Bearing liner thickness, eh Young’s modulus of the bearing liner, Eb Load carrying capacity, W Attitude angle, ) Stationary coordinate system, ( x, y, z ) Displacements of bearing liner in the circumferential, radial and axial directions, uT , u r , u z Angular coordinate, T Deformation coefficient, C d Stiffness matrix for bearing liner, [k ] Elasticity matrix, [ D ] Nodal force vector, {F } Displacement components vector, {d }

constitutive equations. The Gauss-Seidel iterative scheme with over-relaxation is employed to solve the Reynolds equation[15,16]. The boundary conditions are

Fig.4 Distribution of film pressure p

4.1 Pressure field Pressure fields and pressure areas in the bearing are respectively presented in Figs.4 and 5. We note

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that in the EHD theory, the pressure ranges from 45o to 180o while in the HD theory, it covers only about 50°.

Fig.7 Circumferential variation of the film thickness in the median plane

Fig.5 Pressure areas in the bearing

Fig.8 Effect of bearing deformation on minimum film thickness

Fig.9 Effect of deformation coefficient on film geometry

Fig.6 Film geometry

4.2 Film geometry Variation of the oil film thickness is shown in Figs.6 and 7. In the HD theory, the film thickness, according to the circumferential direction, has a sinusoidal shape with a minimum located at T = 180o. In contrast, when the bearing liner is deformed, the film geometry changes and it has two extrema: one is located in the median plane and the other at the edges of the bearing. The increase of film thickness in the middle section and the displacement of its minimum

161

explain the drop of the maximum pressure and the spreading of pressure field compared to the case of HD theory (see Fig.7).

in Fig.8. These results show that the minimum thickness of the film increases with the increasing of C d . For a value of the relative eccentricity H = 0.7 , there is an increase of 57.6 % in the minimum thickness of the film fluid. Figure 9 gives the variation of the fluid film thickness in the circumferential plane

Fig.10 Stiffness coefficients

The change in the minimum thickness of fluid film with the deformation coefficient C d for different values of relative eccentricity İ is presented

Fig.11 Damping coefficients

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for different values of the deformation coefficient. We notice that when the value of C d increases, (1) the thickness of the film in the middle section increases, (2) the point of minimum thickness of the film is moved out of contact. For a value of deformation coefficient equal to 1, the film geometry is altered under the effect of large elastic deformations of the bearing. The fluid film thickness is four times larger than that calculated by considering a rigid bearing (with C d equal to 0). 4.3 Dynamic coefficients Calculation of stiffness and damping coefficients in journal bearing dynamic analysis is performed with respect to the steady-state position. In the case of small displacements of the journal centre, the HD and EHD bearings are characterized by eight dynamic coefficients of stiffness and damping. Figures 10 and 11 show the dynamic coefficients for HD and EHD theories. It is clear that the elastic deformation reduces the stiffness and damping of the bearing. 4.4 Load carrying capacity The effect of deformation on the load carrying capacity is presented in Fig.12. The results show that while the relative eccentricity increses, the value of the load decreases with the increase of the deformation coefficient. The reductions are significant at large deformation. It is considered that the load carrying capacity decreases by 89.5 % for H = 0.2 , 91.6 % for H = 0.4 and 93.8 % for H = 0.7 when C d varies from zero to unity.

Fig.12 Effect of deformation coefficient on load-carrying capacity

4.5 Attitude angle The effect of deformation coefficient on the attitude angle ) is shown in Fig.13. It is observed that, when the relative eccentricity increases, the attitude angle decreases for any values of deformation coefficient.

5. Conclusions The present analysis shows that for the accurate modelling of the bearing behaviour, the elasticity of the shell has to be ultimately considered. Indeed, pressure distribution indicates that peak pressure decreases in the EHD theory. This is due to the deformation of the bearing liner. A similar trend was observed by Sukumaran Nair et al.[9] in their investigation for three-lobe journal bearing. The film pressure area in the HD journal bearing is different from that of an EHD journal bearing, since, the elastic deformation of the bearing tends to enlarge the pressure zone in the bearing.

Fig.13 Effect of relative eccentricity on attitude angle

The variation of static and dynamic characteristics in terms of film geometry, dynamic coefficients, load carrying capacity and attitude angle are computed for both the HD and EHD theories when the bearing operates at different eccentricity ratios. Based on the results thus obtained the followings conclusions are drawn: (1) Appreciable increase in minimum film thickness with the increase in flexibility of bearing liner is also a favourable effect from the designer’s viewpoint. (2) For any eccentricity ratio, stiffness and damping coefficients decreases in the case of EHD journal bearing. (3) Performance characteristics, such as the load capacity and attitude angle, decrease when the bearing liner flexibility increases and these changes are significant especially when the flexible bearing operates at high eccentricity ratio. References [1]

[2]

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