Velocity U

h1h

h2

X

ZPivot point

x

dx

P

Pmax Pressure distribution

X

Let X be the distance from the leading edge, indicating the position of center of pressure

Therefore moment of the load distribution about this point is

(W/L).X =

(pdx is the force acting on a small strip dx)

B

0xdx.p

0 B

Tilting pad bearing – center of pressure

Non-dimensionalizing and writingp as , x as x*B, and W* as , we

get

Or , which on solving gives

*2o

ph

u6 LBU6

h.W2

2o

*1

0

**22o

dxxpB.h

BU6

L

WX

*

1

0

***

W

dxxp

B

X

)15...(K2K2)K1(log)K2(

)K56(K)K1(log)K1)(K3(2

B

X

e

e

It is seen that the value of X/B is determined by the value of K which is determined by the value of h1/h2 = 1+K

Tilting pad bearings – flow rate

The Reynold’s equation for oil flow gives:considering flow only in the x-direction

We know that when dp/dx = 0, h = ho, therefore

Also ho = h2A = ho.2(1+K)/(2+K) where K=(difference of inlet and outlet film thickness)/(outlet film thickness)

Thus

Therefore knowing the inlet and outlet film thicknesses, we can determine the flow rate

dx

dp.

12

h

2

hUq

3

x

2

hUq o

x

)2(

)1(2 K

KUhqx

Tilting pad bearing - friction

The shear stress on each surface is given by the formula (for a Newtonian viscous fluid)

It was earlier shown that

Where U1 is the velocity at z = h and U2 is the velocity at z = 0. In the current discussion of thrust pads, U1 = 0 and U2 is called U

Therefore

z

u

h

UU

2

hz

dx

dp.1

z

u 21

h

U

2

hz

dx

dp.1

z

u

Tilting pad bearing - friction

Therefore stress on fluid layer when z=0 is

Stress when z = h is

The friction F of the pad is the integral in the x and y directions of the shear stress , i.e.

Where L and B are the extent of the pads in the y and x directions respectively

h

U

2

h.

dx

dp0z

h

U

2

h.

dx

dphz

L

0

B

0dxdyF

Tilting pad bearing- friction calculation

• Therefore we get

• Integrating term by term, remembering that and U are constant and h is a function of x only, integrating by parts we get

• p is 0 at x = 0 and x = B. Therefore only the 2nd term in the above equation remains

L

0

B

0

L

0

B

0 0,h0,h dxdyh

U

2

h.

dx

dpdxdyF

L

0

B

0

B

0

L

0

B

0dy

2

dhp

2

hpdxdy

2

h.

dx

dp

Normal load as a function of pressure

• Therefore we have

as h = h2(1+K-Kx/B), and dh= -(h2K/B)dx

• It is obvious that the total load, so

L B L BL B

pdxdyB

Khdxdy

B

Khpdy

dhp

0 0 0 0

22

0 0 222

L

0

B

0Wpdxdy

B

WKhdy

dhp

L B

222

0 0

Tilting pad- friction (contd.)Integrating the 2nd term in the friction expression

We get

Therefore the friction expression becomes

The negative sign in the 1st. Term indicates that the friction force is in a direction opposite to that of the velocity

L

0

B

0

L

0

B

0 0,h0,h dxdyh

U

2

h.

dx

dpdxdyF

L

0

B

0

B

0

B

0o B/KxK1

dx

h

L

h

dxL

h

dxdy

)K1(logKh

LB)B/KxK1(log

Kh

LBe

o

B

0eo

)16...(2

)1(log. 2

20, B

WKh

K

KLB

h

UF eh

DUCOM MICHELL Tilting pad apparatus in the lab

Features–Measurement of pressure distribution along and across the

line of flow–Measurement of temperature at pressure points– Continuously variable sliding speed– Independent gap setting at leading and trailing edge– Oils with different viscosity can be tested to determine the

effect of viscosity on other variables

Therefore it can be used to determine the inter-dependence of the different variables

Picture of apparatus

Manometer tubes showing pressure