OIL FILM THICKNESS IN HYDRODYNAMIC JOURNAL BEARINGS

Hassan Saheed B*(090404024)

DEPARTMENT OF MECHANICAL ENGINEERING, UNIVERSITY OF LAGOS

Email:tolhassan@yahoo.com

ABSTRACT1

The aim of this study is to determine the oil film thickness in real hydrodynamic journal bearings under realistic operating conditions. The study focused on replacing the curved partial bearing with a flat bearing with the aid or Osborne Reynolds equation. Calculations were carried out to determine the oil film thickness and to understand its relationship with other operating parameters. The main test apparatus was a slipper pad Tribology Apparatus.

INTRODUCTION2

Bearings are used to prevent friction between parts during relative movement. In machinery they fall into two primary categories: anti-friction or rolling element bearings and hydrodynamic journal bearings. The primary function of a bearing is to carry load between a rotor and the case with as little wear as possible. This bearing function exists in almost every occurrence of daily life from the watch on your wrist to the automobile you drive to the disk drive in your computer. In industry, the use of journal bearings is specialized for rotating machinery both low and high speed.

Hydrodynamic journal bearings are typical critical power transmission components that carry high loads in different machines. In machine design, therefore, it is essential to know the true or expected operating conditions of the bearings. These operating conditions can be studied both by experimental and mathematical means, for example in test rig experiments, in field or laboratory tests with engines and by calculation or simulation.

OPERATION OF A JOURNAL BEARING3

When the journal is at rest, it is seen from the figure that due to bearing load P, the journal is in contact with the bush at the lower most position and there is no oil film between the bush and the journal. Now when the journal starts rotating, then at low speed condition, with the load P acting, it has a tendency to shift to its sides as shown in the figure. At this equilibrium position, the frictional force will balance the component of bearing load. In order to achieve the equilibrium, the journal orients itself with respect to the bush as shown in figure. The angle θ, shown for low speed condition, is the angle of friction. Normally at this condition either a metal to metal contact or an almost negligible oil film thickness will

prevail. At the higher speed, the equilibrium position shifts and a continuous oil film will be created as indicated in the third figure above. This continuous fluid film has a converging zone, which is shown in the magnified view. It has been established that due to presence of the converging zone or wedge, the fluid film is capable of carrying huge load. If a wedge is taken in isolation, the pressure profile generated due to wedge action will be as shown in the magnified view.

Hence, to build-up a positive pressure in a continuous fluid film, to support a load, a converging zone is necessary. Moreover, simultaneous presence of the converging and diverging zones ensures a fluid film continuity and flow of fluid. The journal bearings operate as per the above stated principle.

FIG 1: OPERATION OF A JOURNAL BEARING

HYDRODYNAMIC THEORY4

The experimental investigations by Petroff and Tower form the background of the hydrodynamic theory. Later on Osborne Reynolds conducted experiments and published the findings in the form of present day hydrodynamic theory of lubrication and the corresponding mathematical equation is known as Reynolds’ equation.

The coefficient of friction obtained by Tower in his investigations on this bearing was quite low, which is now not surprising. After testing this bearing, Tower later drilled a 1.5 diameter lubricator hole through the top but when the apparatus was set in motion, oil flowed out of this hole. In the effort to prevent this, a cork stopper was used, but this popped out and so it was necessary to drive a wooden plug into the hole . When the wooden plug was pushed

out too, Tower, at this point, undoubtedly realized that he was on the verge of discovering. A pressure gauge connected to the hole indicated a pressure of more than twice the unit bearing load.

Tower investigated the bearing film pressure in detail throughout the bearing width and length and reported a distribution similar to that in the figure 2 below, The result obtained by tower had such regularity that Osborne Reynolds concluded that there must be a definite equation relating the friction the pressure and the velocity. The present mathematical theory of lubrication is based upon Reynolds work following the experiment by Tower so the original differential equation developed by Reynolds was used to explain Towers result. Reynolds pictured the lubricant as adhering to both surfaces and being pulled by the moving surface into a narrow wedge-shaped space so as to create a fluid or film pressure of sufficient intensity to support the bearing load.

One of the important assumption simplification resulted from Reynolds’ realization that “the fliud film were so thin in comparison with the bearing radius that the curvature could be neglected”.

FIG 2: APPROXIMATE PRESSURE DISTRIBUTION CURVE OBTAINED BY TOWER

This enabled Reynolds to replace the curved partial bearing with a flat bearing, called the slider bearing, and some of his other he made are:

1. The lubricant obeys Newton’s viscous effect2. The force due to the inertia of the lubricant are negligible3. The lubricant is assumed to be incompressible4. The viscosity is assumed to be constant throughout the film5. The pressure does not vary in the axial direction.Fig 3 shows a journal rotating in the clockwise direction supported by a film of

lubricant of variable thickness h on a partial bearing which is fixed and the journal bearing has a constant surface velocity U

FIG 3: COMPARING A THE CURVED GEOMERTY OF A JOURNAL BEARING (a) WITH A FLAT/SLIDER OR BEARING (b)

Using Reynolds assumption that curvature can be neglected; so we fix a right handed xyz reference system to the stationary bearing and made the following additional assumptions.

6. The bushing and journal extend infinitely in the z direction that is there is no lubricant flow in the z direction.

7. The film pressure is constant in the film depends only on the x coordinate 8. The velocity of any particle of lubricant in the film depends only on the

coordinates x and y. Now an element is selected of lubricant in the film (Fig 3a) of dimensions dx, dy

and dz and compute the forces that act on the sides of this element. As shown in Fig 3b normal forces due to the pressure, act upon the right and left sides of the element and shear forces due to the velocity, act upon the top and bottom sides. Summing the forces in the x direction gives.

∑ F X=Pdydz−( p+ dpdx dx )dydz−τdxdz+(τ+ δτδy dy)dydz=0 …(1)

This reduces to

dpdx

= δτδy

…………………………………………… (2)

From τ=μδ τδ y

…………………………………………………………(3)

Where the partial derivative is used because the velocity U depends upon both x and y, so putting (3) in (2)

dpdx

=μδ2uδ y2 …………………………… …………………………………..(4)

Holding x constant we now integrate this expression twice with respect to y.

δuδy

=1μdpdx

y+C1 …………………………………………………………..(5)

U= 12 μ

dpdx

y2+C1 y+C2 ………………………………………………….(6)

The act of holding x constant means x that C1 and C2 can be functions of x. We now assume that there is no slip between the lubricant and the boundary surfaces. This gives two set of boundary conditions for evaluating the constant C1 and C2:

At y = 0, u = 0 At y = h, u = U

FIG 4: VELOCITY OF LUBRICANT

Substituting these conditions in Eq 6 and solving for the constant gives

C1=Uh

+ h2μ

dpdx

C2=0

Or

u= 12 μ

dpdx

( y2−hy )+Uhy ……………………………………………………………(7)

Fig 4 shows the superposition of these distribution to obtain the velocity for particular values of x and dp/dx

When pressure is maximum, dp/dx = 0 and the velocity is

u=Uhy …………………………………………………………………………..(8)

The volume of the lubricant flowing in the x direction per unit time is Q which is defined as

Q=∫0

h

udy…………………………………………………………………………(9)

Substituting u from equation 7 gives

Q=Uh2

− h3

12 μdpdx

………………………………………………………………………(10)

Assuming the lubricant is incompressible, ThusdQdx

=0

From equation 10 we have

dQdx

=U2dhdx

− ddx ( h3

12μdpdx )=0 …………………………………………………….(11)

Or

ddx ( h3

μdpdx )=6U

dhdx

……………………………………………………………...(12)

Equation 12 is the classical Reynolds equation for one-dimensional flow neglecting flow in z direction.

The formation of oil film is influenced by the following factors:

• The contact surfaces must meet at a slight angle to allow formation of the lubricant wedge.

• The oil viscosity must be high to maintain adequate film thickness to separate the

contacting surfaces at operating speeds.

• The oil must be adhering to the contact surfaces for conveyance into the pressure area to

support the load.

• The oil must be distributing itself completely within the bearing clearance area.

• The operating speed must be sufficient to allow formation and maintenance of the oil film.

• The contact surfaces of bearings and journals must be smooth and free from sharp surfaces

that will disrupt the oil film.

EXPERIMENT APPARATUS AND READINGS5

FIG 5: TRIBOLOGY APPARATUS

From the tribology experiment a correlation can be generated between the geometry of a slipper pad or flat bearing and a journal bearing as done by Osborne Reynolds and from this it is more convenient to obtain parameters to determine the oil film thickness from a flat bearing geometry and then relate it to a journal bearing geometry.

The experimental readings are:

Kinematic viscosity of the lubricant (SAE 40) (v) = 12.5mm2/sSpeed of the pad ω was set to 4.3rads/sec Micrometre screw gauge clearances (ψ) was varied between 1, 1.25 and 2 mm Lubricant temperatures (SAE 40)= 32.5, 33, 33.5 degrees centigrade Density of the lubricant (SAE 40) = 885kg/m3

TABLE 1 When ψ=1mm

When ψ= 1mm

When ψ= 1.25mm

When ψ=1.25mm

When ψ=2mm

When ψ=2mm

Horizontal piezometer reading (mm)

Vertical piezometer reading (mm)

Horizontal piezometer reading (mm)

Vertical piezometer reading (mm)

Horizontal piezometer reading (mm)

Vertical piezometer reading (mm)

125 105 114 98 103 94153 144 143 138 129 130183 154 169 154 152 144179 156 176 157 160 146171 154 172 153 159 140156 143 156 136 143 124106 110 106 100 97 91

DETERMINATION OF OIL FILM THICKNESS6

The minimum oil film thickness is calculated in three phases, In the first phase, the Sommerfeld number So (a dimensionless parameter used in bearing performance calculations) is determined approximately by the following equation, based on the measurement data:

So= F φ2

DBvρω

Where F is the bearing load Ψ is the relative bearing clearance D is the diameter of the bearing

B is the width of the bearingη is the dynamic viscosity, ν is the kinematic viscosity,

ρ is the density, ω is the hydrodynamic angular velocity.

Calculating the sommerfeld So number as the Relative Clearance and Load on the Journal Bearing varies.

TABLE 2

Ψ (mm)

F (N) D (mm)

B (mm)

v (mm2/s)

ρ (kg /mm3)

ω (rads/s)So= F φ2

DBvρω

1 14 0.32 0.85 12.5 0.885 4.3 1.0821.25 17.5 0.32 0.85 12.5 0.885 4.3 1.3532 28 0.32 0.85 12.5 0.885 4.3 2.164

In the second phase, the relative eccentricity ε was determined approximately as a function of the Sommerfeld number So and the width-to diameter ratio DB of the bearing (see Table 2). The approximation was made for a plain bearing with the width-to-diameter ratio DB = 0.32 mm / 0.85 mm = 0.376, and the relative eccentricity ε was calculated by the following approximate equation:

ε ≈ K1Sok2

The values of the coefficients K1 and k 2 in the equation above are presented in Table 2

TABLE 3K1 K2

1≤So<10 0.798 0.07310≤So<100 0.897 0.022100 ≤So<¿ 200 0.980 0.0028

Calculating eccentricity ε of the Journal bearing;

TABLE 4Ψ (mm)

So= F φ2

DBvρωε ≈ K1So

k2

1 1.082 0.80261.25 1.353 0.81582 2.164 0.8443

In the third phase, the following equation was used to calculate the minimum oil film thickness h0 as a function of the bearing diameter, relative bearing clearance and relative eccentricity:

h0=12DΨ (1−ε)

Where D is the diameter of the bearing, ψ is the relative bearing clearance,

ε is the relative eccentricity.

Calculating the oil film thickness h0

TABLE 5Ψ (mm) D (mm) ε ≈ K1So

k2

h0=12DΨ (1−ε)

1 0.32 0.8026 0.03161.25 0.32 0.8158 0.03682 0.32 0.8443 0.0500

INDUSTRIAL APPLICATIONS7

The journal bearing has several advantages over other types of bearings, provided that it has a

constant supply of clean high-grade motor oil. First, it handles high loads and velocities

because metal-to-metal contact is minimal due to the oil film. These bearings are also

remarkably durable and long lasting, and because of the damping effects of the oil film, they

may also help make engines quiet and smooth running. In part because of these inherent

advantages this sort of bearing is used in more than just gas and diesel-fuelled piston engines.

It tends to be common in many high-loads, high-velocity applications, including a range of

industrial machines and turbines.

CONCLUSION8

1. The technology of lubrication has been used from the ancient times, from the pyramid building where massive rock slabs are moved, up to present modern times.

2. The main purpose of lubrication is to reduce friction and wear in bearings or slidingcomponents to prevent premature failure.

3. Adequate lubrication also helps to prevent foreign material from entering the bearings and guards against corrosion and rusting. Satisfactory bearing performance can be achieved by adopting the lubricating method that is most suitable for the particular application and operating conditions

4. A lubricant prevents the direct contact of rubbing surfaces and thus reduces wear. It keeps the surface of metals clean. Lubricants can also act as coolants by removing heat effects and also prevent rusting and deposition of solids on close fitting parts.

REFERENCES9

[1] Rudnick Leslie R., Ewa A. Bardasz, and Gordon D. Lamb; "Lubricant Additives: Chemistry and Applications", Marcel Dekker, pages 387-427, (2003).

[2] Stachowiak Gwidon W.,and Andrew W. Batchelor; Engineering Tribology", third ed‐ ition, Amsterdam: Elsevier, pages 2,12,-22,52,62-67,77, (2005).

[3] Geore J.W.; "Lubrication Fundamentals", (1980).

[4] Dowson D.; History of Tribology, 2nd Edition, Professional Engineering Publishing, London (1998).

[5] J.E Shigley and C.R Mischke , Mechanical Engineering Design , McGraw Hill Publication, 5th Edition. 1989. [6] M.F Spotts, Design of Machine Elements, Prentice Hall India Pvt. Limited, 6th Edition, 1991.