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).

142

(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 of a journal. Since, the pressure is created within the system due to rotation of the shaft, this type of bearing is known as self acting bearing. The pressure generated supports the external load W. This mode of lubrication is seen in bearings mounted on engines and centrifugal pumps.

143

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.

Figure 4 Boundary Lubrication

144

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.

145

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

2U N 'π= and F Aτ= (1) 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

Uh

τ µ= (2)

or 2 rNh

'πτ µ= (2a)

Hence,

2 24 'N L rF

Cµ π

= (3)

Further, the frictional torque may be obtained as

146

2 34 '.fN L rT F r

Cµ π

= = (4)

This equation is known as the Petroff’s equation, for lightly loaded journal bearings. The coefficient of friction may be obtained as

FfW

= (5)

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

2WPrL

= (6)

Hence, the coefficient of frictional is

2 '2 N rfP C

µπ= (7)

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.

147

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

148

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).

149

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. τx is the shear stress along the x direction while p is the fluid film pressure.

150

Figure 12 Converging Oil Film

Figure13 Infinitesimal Element in Equilibrium

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

0xx

ddppdydz p dx dydz dxdz dy dxdzdx dy

ττ τ⎛ ⎞⎛ ⎞− + + − + =⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ (9)

xddpdx dy

τ⇒ = − (10)

From Newton’s viscous flow, we have

xdudy

τ µ= − (11)

where u is the velocity in the x-direction. Hence,

2

2

dp d u d udx dy y dy

µ µ⎛ ⎞∂

= − − =⎜ ⎟∂⎝ ⎠ (12)

151

or 2

2

1d u dpdy dxµ

= (13)

On integrating we get,

11du dp y C

dy dxµ= + (14)

21

12

dpu y C ydxµ

= + 2C+ (15)

The boundary conditions are: At y = 0, u = U (16.1) At y = h, u = 0 (16.2) On applying the boundary conditions, we obtain the constants as

C1 = U (17)

and 21

2dp UC hdx hµ

= − − (18)

Hence,

( )212

dp h yu y hy Udx hµ

−= − − + (19)

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 uu dx dydz v dy dxdzx y

⎡ ⎤∂ ∂⎡ ⎤+ + +⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦ (21)

On applying the conservation of mass, we get vy x∂ ∂u

= −∂ ∂

(22)

Substituting the value of u and on rearranging, we get

( ) ( )21 U h ydpdv y hy dyx dx hµ

−⎡ ⎤∂= − − +⎢ ⎥∂ ⎣ ⎦

(23)

( ) ( )2

0 0

1 0y h y h

y y

U h ydpdv y hy dyx dx hµ

= =

= =

⎛ ⎞−⎡ ⎤∂= − − + =⎜ ⎟⎢⎜ ⎟∂ ⎣ ⎦⎝ ⎠

∫ ∫ ⎥ (24)

On simplifying we get, 3

012 2

d dp h d Uhdx dx dxµ

⎡ ⎤ ⎡ ⎤− =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ (25)

152

The above equation can be rearranged as

3 6d dp dh Udx dx dx

µ⎡ ⎤ =⎢ ⎥⎣ ⎦h (26)

The equation represents the Reynolds’ equation in two dimensions, expressing the pressure gradient in a converging oil film as a function of film thickness, viscosity of the 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

153

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 21

h hh h xL−

= − (27)

Defining some non-dimensional terns

1 2h hL

α −= , 2ha

L= and xX

L= (28)

Hence, the expression for the thickness of the oil film at any cross-section can be re-expressed as

1h LX hα= + (29)

But 1h La Lα= − (30)

Hence, the oil film thickness can be expressed as

( )h L X aα α= + − (31)

From Reynolds’ equation, we have

3 6d dp dh Udx dx dx

µ⎡ ⎤ =⎢ ⎥⎣ ⎦h (26)

On integration, we get

316dph Uh

dxµ C= + (32)

or 2 3

16dp BUdx h h

µ ⎡ ⎤= +⎢ ⎥⎣ ⎦ (33)

where 1

6BB

Uµ=

From equations (28), (31) and (33), we get

154

( ) ( )2

6 1dp U Bdx L a X L a X

µα α α α

⎡ ⎤= +⎢ ⎥

− + − +⎢ ⎥⎣ ⎦2 (34)

or ( ) ( )2

6 U dX BdXdpL a X L a Xµ

α α α α

⎡ ⎤= +⎢ ⎥

− + − +⎢ ⎥⎣ ⎦2 (35)

On integrating equation (35), we get

( ) ( )26

2U dX BdX

2p CL a X L a Xµ

α α α α α

⎡ ⎤= − + +⎢ ⎥

− + − +⎢ ⎥⎣ ⎦ (36)

The boundary conditions are:

At X = 0, p = 0 (37.1) At X = 1, p = 0 (37.2)

On substituting equation (29) in equation, we get

( )62

UCL a

µα α

=−

(38)

22

aB Laaα

α−

=−

(39)

Hence, the pressure distribution along the idealized plane slider bearing can be expressed as

( )( )( )2

6 12

X XUpL a a X

αµα α α

⎡ ⎤−= ⎢

− − +⎢ ⎥⎣ ⎦⎥ (40)

The load carrying capacity can be expressed as

0

L

W wpdx= ∫ (41)

1

0

W wLpdX= ∫ (42)

On substituting the value of p from equation (40), we get

( )1 2

20

62

X XW wU dXa a X

αµα α α

−=

− − +∫ (43)

On integrating, the load carrying capacity can be expressed as

2

26 ln2

w a aW Ua aαµ

α α⎡ −⎛ ⎞= ⎜ ⎟

⎤+⎢ ⎥−⎝ ⎠⎣ ⎦

(44)

155

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

1

n

i ii

F Aτ=

= ∑ or F dAτ= ∫

Now, from Newton’s law, we have

xdudy

τ µ= − (11)

where u is the velocity, which can be expressed as

( )212

dp h yu y hy Udx hµ

−= − − + (19)

Differentiating equation (19) w.r.t. y, we get 1 2

2du dp y h Udy dx hµ

−⎡ ⎤= −⎢ ⎥⎣ ⎦ (45)

On substituting LdX = x and equation (45) in equation (11), we get 1 2

2xdp h y u U

L dx hτ −⎡ ⎤= ⎢ ⎥⎣ ⎦

+ (46)

On differentiating equation (40) w.r.t. X and simplifying, the pressure gradient obtained is

( )( )( )3

262

a X aXdp UdX L a a X

α α αµα α α

⎡ ⎤− + −= ⎢ ⎥

− − +⎢ ⎥⎣ ⎦ (47)

Hence, from equations (45), (47) and (11), the shear force acting at any point is

( )( )( )

( )( )3

3 2 2 12x

a X aX a X yUL La a X

α α α α αµτα αα α α

⎡ ⎤⎛ ⎞− + − − + −⎛ ⎞⎢ ⎥= +⎜ ⎟⎜ ⎟⎜ ⎟ − +− − +⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦a X

(48)

On the moving surface y = 0, and hence

[ ] ( )( )( ) ( )0 20

3 2 12x y

a X aXUL aa a X

α α αµτ τα αα α α= X

⎡ ⎤⎛ ⎞− + −⎢ ⎥= = +⎜ ⎟⎜ ⎟ − +− − +⎢ ⎥⎝ ⎠⎣ ⎦

(49)

Hence, the total frictional force F0 acting on the moving surface is 1

0 0 00 0

L

F w dx dxτ τ= =∫ ∫ (50)

On substituting equation (49) in equation (50) and integrating, we get

04 ln

2aF Lw

a aαµ 6

α α⎡ −⎛ ⎞ ⎛= − +⎜ ⎟ ⎜

⎤⎞⎟⎢ ⎥−⎝ ⎠ ⎝ ⎠⎣ ⎦

(51)

The coefficient of friction, f is

156

0

2

4 6ln2

6 2ln2

aLwa aFf

W Uw aa a

αµα

µ α αα α

α⎡ − ⎤⎛ ⎞ ⎛− +⎜ ⎟ ⎜

⎞⎟⎢ ⎥−⎝ ⎠ ⎝ ⎠⎣ ⎦= =

⎡ − ⎤⎛ ⎞ +⎜ ⎟⎢ ⎥−⎝ ⎠⎣ ⎦

(52)

or ( )

( )

22 2 ln 3

3 2 ln 6

aaaf

aaa

αα α α

αα α

−⎛ ⎞− −⎜ ⎟⎝ ⎠=−⎛ ⎞− +⎜ ⎟

⎝ ⎠

(53)

Also, the resultant pressure distribution obtained must be equal and opposite to the vertically applied load. Using the Varigon’s theorem to determine the location of the centre of pressure,

12

0 0

L

lW xwpdx L w pXdX= =∫ ∫ (54)

or ( )( )

21

20

162

X XlW ULw dX

a a Xαµ

α α α

−⎛ ⎞= ⎜ ⎟−⎝ ⎠ − +∫ (55)

or ( )( )

2

4

53 ln 36 2

2

aa a aalW ULw

a

α αα α ααµ

α α

⎡ ⎤−⎛ ⎞− − − +⎜ ⎟⎢ ⎥⎛ ⎞ ⎝ ⎠⎢ ⎥= ⎜ ⎟− ⎢ ⎥⎝ ⎠⎢ ⎥⎣ ⎦

(56)

or ( )( )

( )

2

2

53 ln 32

2 ln 2

aa a aal L

aaa

α αα α α

αα α α

⎡ ⎤−⎛ ⎞− − − +⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥=−⎛ ⎞⎢ ⎥− −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(57)

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.

157

Figure16 Location of the Pivot Point of the Shoe

Consider the following equation

1

2

1hrh a

α= − = − (58)

2h rL

α⇒ = (59)

Using equation (59), the equations for the performance of the plane slider bearing with pivoted shoe can be obtained as

( ) ( )2

2 22

6 1 2ln 12

UwLW rh r r r

µ ⎡ ⎤= + −⎢ ⎥+⎣ ⎦

(44a.1)

or ( )2

22

6w

UwLW gh

µ= r (44a.2)

158

where ( ) ( ) ( )2

1 2ln 12wg r r

r r r⎡ ⎤

= + −⎢ ⎥+⎣ ⎦

( )02

4 ln 12

UwLF rh r r

µ ⎡= + −⎢6 ⎤

⎥+⎣ ⎦ (51a.1)

or ( )00

2F

UwLF g rh

µ= (51a.2)

where ( ) ( )0

4 6ln 12Fg r r

r r⎡ ⎤= + −⎢ ⎥+⎣ ⎦

( )( ) ( )

( ) ( ) 2

51 3 ln 1 32

2 ln 1 2

r r r r rl L

r r r r

⎡ ⎤⎛ ⎞+ + + − +⎜ ⎟⎢ ⎥⎝⎢ ⎥=+ + −⎢ ⎥

⎢ ⎥⎣ ⎦

⎠ (57a)

( )( )

02 16

F

w

g rhfL g r⎡ ⎤

= ⎢ ⎥⎣ ⎦

(60)

( )2f

hf g rL

= (60a)

The shoe has tangential as wall as radial degrees of freedom. The friction force at the pivot junction , balances with fluid friction of the shoe. Moreover, the friction force

and the normal reaction together balance the vertical thrust. Thus, the value of r changes, automatically, to meet the equilibrium conditions.

pF

PRESSURE DISTRIBUTION IN JOURNAL BEARING: Consider a full journal bearing as shown in figure below.

159

Figure 17: Idealized Full Journal Bearing

Figure.18 Unwrapped Film

Let Radius of the journal = r Radius of the bearing = r + c Clearance = c Eccentricity = e Vertical Load = W Angle of Inclination of the vertical load = α

h = CA – CB = r + c – CB (61) From and , we have CBE∆ 'C BE∆

sinsin sinBE rCB φθ θ

= = (62)

Since, φ θ β= −

160

(sin cos cos sinsin

rCB )θ β θθ

= − β (62a)

From and , we have 'C DC∆ 'C BD∆sin sine rθ β= (63)

22

2cos 1 siner

β θ= − (63a)

On substituting the values of sin β and cos β from equations (63) and (63a) in equation (62a), we get

2 2 2sin cosCB r e eθ θ= − − (64) Substituting equation (64) in equation (61), the final expression for film thickness as a function of θ is obtained as

2 2 2sin cosh r c r e eθ θ= + − − + (65)

On neglecting 2 2sine θ in comparison with r2, and further using

enc

= (66)

where n is called the eccentricity ratio, the equation (65) can be simplified as ( )1 cosh c n θ= + (67)

On substituting x rθ= in Reynold’s equation, we get

3 6d dp dh Ud d d

µ hθ θ θ⎡ ⎤ =⎢ ⎥⎣ ⎦

(26a)

On integrating, we get

3 6 6dp dhh Ur d Urh Kd d

µ θ µθ θ= =∫ + (68)

or ( ) ( )

122

6 11 cos 1 cos

Kdp Urd c n c n

µθ θ θ 3

⎡ ⎤= +⎢ ⎥

+ +⎢ ⎥⎣ ⎦ (69)

or ( ) ( )

122

61 cos 1 cos

K dUr dpc n c n

θµ θθ θ 3

⎡ ⎤= +⎢ ⎥

+ +⎢ ⎥⎣ ⎦∫ (70)

Further, we have

0 2 2 0 0p p p pθ θ π θ π θ= = = == ⇒ − = (71)

which gives

( ) ( )

21

2 320

6 1 01 cos 1 cos

KUrp dc n c n

θ π

θ

µ θθ θ

=

=

⎡ ⎤= +⎢ ⎥

+ +⎢ ⎥⎣ ⎦∫ = (72)

161

or ( )

( )

2

201

2

30

1 cos

1 cos

dnK

c dn

θ π

θθ π

θ

θθ

θθ

=

==

=

+= −

+

∫

∫ (73)

On simplifying, we get

( )2

1 2

2 12

c nK

n−

=+

(74)

From equation (69), it can be easily observed that 0dpdθ

= when ( )1 1 cosK h c n θ= − = − +

and the minimum film thickness hm at minimum and maximum pressure is obtained as

( ) ( ) ( )2

1 2max min

2 12m p p

c nh h h K

n= =

−= = = − =

+ (75)

Correspondingly, the value of θ where the maximum and minimum pressured occur is given by

2

3cos2

nn

θ −=

+ (76)

Substituting the value of K1 in equation (72), the pressure at any angle θ from the radial line is given by

( ) ( )0 220

6 11 cos 1 cos

mhUr3p p d

c n c n

θ

θµ θ

θ θ

⎡ ⎤= + −⎢ ⎥

+ +⎢ ⎥⎣ ⎦∫ (77)

or ( ) ( )

2

0 2 320

6 1 1cos cos

mh AUrAp p dc cA A

θ

θµ θ

θ θ

⎡ ⎤− = −⎢ ⎥

+ +⎢ ⎥⎣ ⎦∫ (78)

On solving, we get

( )( )( )0 22 2

2 cos sin62 1 cos

n nUrp pc n nθ

θ θµθ

⎡ ⎤+⎢ ⎥− =⎢ ⎥+ +⎣ ⎦

(79)

162

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

sin sin 0Lrp d Wπ

θ θ θ α− =∫ (80)

Since, for an idealized bearing, α = 900, the load carrying capacity is given by

2

0

sinW Lr p dπ

θ θ θ= ∫ (81)

Substituting the value for pθ and solving, we get

( )

2

2 2 2

122 1r ULnW

c n nπµ

=+ −

(82)

Choosing '

2UN

rπ= and

2WPLr

= , we get

( )2 22 '

2

2 112

n nr Nc P n

µπ

+ −⎛ ⎞ =⎜ ⎟⎝ ⎠

(83)

The quantity 2 'r N

c Pµ⎛ ⎞

⎜ ⎟⎝ ⎠

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

eccentricity ratio (figure 20).

163

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

1 22

du dp y h Udy dx hµ

−⎡ ⎤= −⎢ ⎥⎣ ⎦

Now, y = 0 represents the journal surface. Substituting for 0y

uy =

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

in the previous

equation, we get

( ) 0 2J x y

U h dph d

τ τ µ=

= = +x

(84)

Substituting for x rθ= and dx rdθ= , we get

2JU h dph r dµτ

θ= + (85)

164

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

from

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

( )( )

( )( )

2

22

6 141 cos 2 1 cosJ

nUc n n nµτ

θ θ

⎡ ⎤−⎢ ⎥= −

+⎢ ⎥+ +⎣ ⎦ (86)

The total frictional force may be obtained as

( )( )

( )( )

22

220

6 141 cos 2 1 cosJ J

nUF d Lrc n n n

πµ dτ θ θθ θ

⎡ ⎤−⎢ ⎥= = −

+⎢ ⎥+ +⎣ ⎦∫ ∫ (87)

On integrating and simplifying, we get

( )( )

2

2 2

4 1 2

2 1J

nULrFc n n

πµ +=

+ − (88)

The non-dimensional form of the equation can be expressed as

( )( )

2 2'

2

2 1

4 1 2f

n nr Nc P nµ

π

+ −=

+ (89)

where '2U Nπ= and 2

Jf

FPrL

= .

165

Figure 21 rc

(Horizontal Line represents rcµ

From figure 21, it can be observ

Sommerfeld number greater than 0.

JF is independent of n for Sommerf

4

JF =

(90) Equation (90) is similar to the Petrois neglected), applicable for lightly l Substituting n = 0 in the above equa

Sommerfeld Number

'

f

NPµ v/s Sommerfeld Number

'

2

1 0.05062f

NP π

= ≈ , for lightly loaded bearings)

ed that the value for '

f

r Nc Pµ remains constant for

15. In other words, the value of the Frictional Force eld values greater than 0.15. Hence,

2 2Lr Nc

π µ '

for 0.15S >

ff’s equation (while Petroff’s equation, the effect of n oaded bearings.

tion, we get

166

'

2

1 0.05062f

r Nc Pµ

π= =

For an idealized journal bearing, the coefficient of friction can be found as

( )( )

( )

2

2 2 2

2

2 2 2

4 1 2

2 1 1 212 3

2 1

J

nULrc n nF c nf

r ULnW rc n n

πµ

πµ

+

+ −

n⎡ ⎤+

= = = ⎢ ⎥⎣ ⎦

+ −

(91)

Rearranging the above equation, 21 2

3r fc n

n⎡ ⎤+= ⎢ ⎥⎣ ⎦

(92)

The equation (92) shows that r fc

is a function of n only.

Now, multiplying both sides of equation (7), the relationship of r fc

with Sommerfeld

number of a lightly loaded bearing can be obtained as

2

2 '2r rfc c

µπ NP

⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(93)

Exercise problems 1. Obtain the Pressure distribution (p v/s x) plot and determine the maximum pressure developed for a plane slider bearing with the following data: 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

167

168