Chapter_8_Dynamic analysis of fydrodynamic bearings (1)

CHAPTER 8 DYNAMIC ANALYSIS OF HYDRODYNAMIC BEARING In this chapter the analyses of the hydrodynamic bearings such as plane slider bearing and journal bearing are discussed.. The derivati

CHAPTER 8 DYNAMIC ANALYSIS OF HYDRODYNAMIC BEARING

In this chapter the analyses of the hydrodynamic bearings such as plane slider bearing and journal bearing are discussed Briefly different types of lubrications are described and the mechanism of pressure development in the oil film is studied The Petroff’s equation for a lightly loaded journal bearing is derived The derivation of Reynold’s equation is carried out and it is applied to idealized plane slider bearing with fixed and pivoted shoe and journal bearings

Lubrication

Lubrication is the science of reducing friction by application of a suitable substance called lubricant, between the rubbing surfaces of bodies having relative motion The main motive of using a lubricant is to reduce friction, to reduce or prevent wear and tear, to carry away heat generated in friction and to protect against corrosion The basic modes of lubrication are thick and thin film lubrication

Thick Film Lubrication:

Thick film lubrication describes a condition of lubrication, where two surfaces of bearing

in relative motion are completely separated by a film of fluid Since there is no contact between the surfaces, the properties of surface have little or no influence on the performance of the bearing The resistance to the relative motion arises from the viscous resistance of the fluid Therefore, the performance of the bearing is only affected by the viscosity of the lubricant Thick film lubrication is further divided into two groups: hydrodynamic and hydrostatic lubrication

Hydrodynamic Bearing: Hydrodynamic lubrication is defined as a system of lubrication

in which the supporting fluid film is created by the shape and relative motion of the sliding surfaces

The principal of hydrodynamic bearing is shown in fig.1 Initially the shaft is at rest (a)

and it sinks to the bottom of the clearance space under the action of load W As the

journal starts to rotate, it will climb the bearing surface (b) and as the speed is further increased, it will force the fluid into the wedge-shaped region (c)

(a) (b) (c)

Figure 1 Formation of Continuous Film in a Journal Bearing

Figure 2 Hydrodynamic Lubrication (Oil Wedge Region)

Since more and more fluid is forced into the wedge-shaped clearance space, pressure is generated within the system Fig.3 shows the pressure distribution around the periphery

Figure 3 Pressure Distribution in Hydrodynamic Bearing

Hydrostatic Lubrication: Hydrostatic lubrication is defined as a system of lubrication in

which the load supporting fluid film, separating the two surfaces, is created by an external source, like a pump, supplying sufficient fluid under pressure Since the lubricant is supplied under pressure, this type of bearing is called externally pressurized bearing Hydrostatic bearings are used on vertical turbo-generators, centrifuges and ball mills

Thin Film Lubrication:

Thin fluid lubrication, also known as boundary lubrication, is defined as a condition of lubrication, where the lubricant film is relatively thin and there is partial metal to metal contact This mode of lubrication is seen in door hinges and machine tool slides The conditions of boundary lubrication are excessive load, insufficient surface area or oil supply, low speed and misalignment

The hydrodynamic bearing also operates under the boundary lubrication condition when the speed is very low or when the load is excessive

Under the extreme conditions of load and temperature, the fluid film gets completely ruptured, direct contact between the two metallic surfaces takes place and thus, extreme boundary lubrication exists

Figure 5 Contacts at High Points (Extreme Boundary Lubrication)

The phenomenon of extreme boundary lubrication is based on the theory of hot spots

These hot spots, also known as high spots are the spots on the metallic surfaces where the welding of the two surfaces takes place, owing to extreme temperature conditions, which

is a consequence of the shearing action of the high points However, due to the relative

motion between the two surfaces, the welding too gets ruptured As a consequence of the phenomenon of the high spots, occurring at extreme conditions of load and temperature, the physical properties get severely damaged

LIGHTLY LOADED JOURNAL BEARINGS:

The following assumptions are made while deriving the characteristic equations for the lightly loaded journal bearings:

1 The radial load is almost zero

2 Viscosity of the lubricant is very high

3 Journal speed approaches very large values

4 Film thickness is very small as compared to radius of the journal i.e h <<< r

Figure 6 Journal Bearing

Figure 7: Unwrapped Film

Fig.7 shows the unwrapped film The length is 2πr and the width is L into the plane of the paper Also, the film thickness is equal to the clearance i.e h = C

Now, we have

2

where N’ = journal speed

τ = shear stress acting on the fluid

A = 2πrL, area of the journal surface

Assuming constant coefficient of viscosity of the fluid and from Newton’s law, we have

U h

2 3

4 '

This equation is known as the Petroff’s equation, for lightly loaded journal bearings

The coefficient of friction may be obtained as

F f W

We define unit bearing load P as the radial load per unit projected area

2

W P rL

PRESSURE DEVELOPMENT IN THE OIL FILM:

Consider two parallel surfaces, one stationary and the other moving with uniform

velocity U, as shown in fig.8

Figure 8 Two Parallel Surfaces in Motion

Here, we assume that the two surfaces are very large in a direction perpendicular to the plane of motion and therefore, their velocity in this direction is zero Since, the velocity

of the oil film varies uniformly from zero at the stationary surface ST to U at the moving surface MN, therefore, the pressure developed in the oil film is zero That is, the moving

surface cannot take any vertical load and even a small load is applied, the oil film will squeeze out

Consider another case similar to the previous case, the only difference being that here the direction of motion of the moving surface is vertical and not horizontal Due to the

motion of the surface MN, oil film is squeezed out and the velocity increases from zero at the central section CC1 to a maximum at the outlet sections MS and NT The distribution

of velocity is shown below

Figure 9 Two Parallel Surfaces, One Stationary and the Other in Vertical Motion

We observe from the figure that the maximum velocities occur at the midpoints for each cross-section This type of velocity distribution occurs only if the maximum pressure is at the central cross-section CC1, falling out to zero value at the outlet cross-sections MC and

NT Such a kind of flow is known as pressure induced flow

Lastly, consider another case similar to the first case, the only difference being that the stationary surface here is inclined at an angle α to the line of motion

Figure10 Stationary Surface Inclined at an angle α to the Line of Motion

The velocity distribution of the oil film is shown in fig.11(a)

Figure11 (a) Velocity Distribution of the Oil Film

Considering only unit thickness perpendicular into the plane of paper Volume of fluid entering the space is given by SMO and that leaving is given by NPT, with MO and NP representing the velocities at the moving surface Since some vertical load is applied, therefore some amount of fluid is squeezed out of the space between the two plates The velocity distribution due to this pressure induced flow is shown in fig.11(b)

Figure11 (b) Velocity Distribution of the Oil Film

Fig.11(c) shows the resultant velocity distribution, thereby balancing the volume of fluid entering and leaving the space between the two surfaces Also, owing to pressure induced flow, pressure is developed in the oil film with a maximum value at the cross-section CC1, where such a flow is zero, as at the outer sections MS and NT The pressure distribution is also shown in fig.11(c)

Figure11 (c) Velocity Distribution of the Oil Film

DERIVATION OF REYNOLD’S EQUATION:

The theory of hydrodynamic bearing is based on a differential equation derived by

Osborne Reynold Reynold’s equation is based on the following assumptions:

1 The lubricant obeys Newton’s law of viscosity

2 The lubricant is incompressible

3 The inertia forces of the oil film are negligible

4 The viscosity of the lubricant is constant

5 The effect of curvature of the film with respect to film thickness is neglected It is assumed that the film is so thin that the pressure is constant across the film thickness

6 The shaft and bearing are rigid

7 There is a continuous supply of lubricant

An infinitesimally small element having dimensions dx, dy and dz is considered in the analysis u and v are the velocities in x and y direction τ x is the shear stress along the x

direction while p is the fluid film pressure

Figure 12 Converging Oil Film

Figure13 Infinitesimal Element in Equilibrium

On balancing the force acting in the x-direction, we get

0

x x

d dp

where u is the velocity in the x-direction

Hence,

2 2

dx = −dy⎛⎜−µ∂y⎞⎟=µ dy

∂

or 2 21

d u dp

On integrating we get,

11

du dp

y C

2 1

12

Now, considering the flow between the two surfaces ST and MN, where the distribution

of velocity for a section AB is represented

Volume of fluid entering the element = udydz + vdxdz (20)

Volume of fluid discharging = u u dx dydz v u dy dxdz

The above equation can be rearranged as

lubricant and the relative velocity of the moving surface

IDEALIZED PLANE SLIDER BEARING (Fixed Shoe):

Consider a plane slider bearing with a fixed shoe

Figure 14 Plane Slider Bearing With Fixed Shoe of Length ‘L’

Figure 15 Film Thickness and Inclination Angle

Let

Length of the shoe = L

Surface velocity (uniform) = U

Force acting = F

External load acting vertically = W

Width of the moving surface = w

Thickness of the film (at entrance) = h1

Thickness of the film (at exit) = h2

Angle between the fixed shoe and the x-axis = α

The thickness of the oil film at any distance can be expressed as

1 2 1

a

αα

αµ

62

X X

αµ

For calculating the total frictional force acting on the moving surface, the shear forces acting on the elemental areas need to be determined

F f

αµ

a f

a a

1 2

2 0

16

IDEALIZED PLANE SLIDER BEARING (Pivoted Shoe):

The principal characteristics of a plane slider bearing depend on the geometry of the bearing, lubricant viscosity and the speed of the moving member

In case of a plane slider bearing with fixed shoe, if the load increases beyond the capacity, the bearing may cease to operate under hydrodynamic conditions To improve the performance of the bearing under such conditions i.e to improve the stability of the bearing, the normal practice is to pivot the shoe so that the inclination of the fixed member is changed automatically to suit the load conditions Moreover, the difficulty in manufacturing a very thin fluid film in the plane slider bearing is also overcome

Figure16 Location of the Pivot Point of the Shoe

Consider the following equation

1 2

1

h r

F w

h f

and the normal reaction together balance the vertical thrust Thus, the value of r changes,

automatically, to meet the equilibrium conditions

p

F

PRESSURE DISTRIBUTION IN JOURNAL BEARING:

Consider a full journal bearing as shown in figure below

Figure 17: Idealized Full Journal Bearing

Figure.18 Unwrapped Film

Let

Radius of the journal = r

Radius of the bearing = r + c

(sin cos cos sinsin

0

K Ur

θ π θ

or ( )

2

2 0

1

2

3 0

d n K

θθθθ

2

n n

3

θ θ

θ θ

Figure 19 Pressure Distribution in the Film of a Journal

CHARACTERISTIC OF JOURNAL BEARING:

Considering the equilibrium of forces in y-direction, we get

2 0

⎝ ⎠ is called the Sommerfeld number, which is a function of the

eccentricity ratio (figure 20)

Figure 20 Eccentricity Ratio (n) v/s Sommerfeld Number The frictional force on the moving surface can be determined by the summation of the elementary shear stresses on the Journal surface

From equation (53), we have

θ

Substituting the values of the film thickness h and the pressure gradient dp

dθ from equations (75) and (78), the frictional force on the Journal surface τJ is obtained as

2 2 2

6 14

J

n U

µτ

2 2

0

6 14

n U

πµ

F P rL

Sommerfeld NumberFigure 21

(Horizontal Line represents

' 2

10.05062

= ≈ , for lightly loaded bearings)

From figure 21, it can be observed that the value for

Sommerfeld number greater than 0.15 In other words, the value of the Frictional Force

J

F is independent of n for Sommerfeld values greater than 0.15 Hence,

2 24

J

Lr N F

Equation (90) is similar to the Petroff’s equation (while Petroff’s equation, the effect of n

is neglected), applicable for lightly loaded bearings

Substituting n = 0 in the above equation, we get

' 2

10.05062

r f

Length of the Bearing 10cm

Width of the Bearing = 6 cm

Velocity =4 m/s

Viscosity of the lubricant = 100 cp

Minimum Fluid Film Thickness = 0.002 cm

Maximum Fluid Film Thickness = 0.006 cm

2 In a journal bearing, diameter of the bearing = 3 cm, length of the bearing = 6 cm, speed =

2000 rpm, radial clearance = 0.002 cm, inlet pressure 0.3 Mpa Location of the inlet hole = 300 , viscosity = 25 cp, eccentricity ratio = 0.1 Radial load = 500 N and Sommerfeld number is calculated to be 0.1688 Find friction torque on the journal, coefficient of friction and power loss and load carrying capacity.

0