Page 367 - Analysis and Design of Machine Elements

(2) Load carrying capacity coefficient Sliding Bearings 345
Convert variables in an xyz rectangular coordinate system to a polar-cylindrical coor-
dinate system with z axis along the axis of journal, we have
dx = rd

v = r
Substitute these relations and Eq. (12.16), Eq. (12.17) into Eq. (12.10) and the Reynolds
equation in a cylindrical coordinate system is expressed as
dp (cos − cos )
0
= 6 ⋅ ⋅ (12.18)
d 2 (1 + cos ) 3
Integrating from to where pressure is created in the bearing, we have
1

(cos − cos )
0
p = 6 d (12.19)

2
∫ (1 + cos ) 3
1
Integrating from to , the total load in the radial load direction for unit width is
1 2
2 ∘
p = p cos[180 −( + )]rd
y

a
∫
1
[ ]
r 2 (cos − cos ) ∘
0
= 6 d cos[180 −( + )] d (12.20)
a
2
∫ ∫ (1 + cos ) 3
1 1

The circumferential pressure distribution in a hydrodynamically lubricated bearing is
illustrated in Figure 12.8b. The pressure increases gradually as the rotating journal draws
viscous oil into the converging wedge, approaching the maximum value and falls rapidly
as the space between the journal and bearing diverges again. Equation (12.20) expresses
the pressure in the external load direction in one cross section of the journal bearing.
Neglecting side leakage, the integrated effect of the pressure distribution is a force
sufficient to balance the applied load and support the shaft on the oil film without
metal-to-metal contact [3]. Therefore, for a limited width sliding bearing, the total load
capacity is
[ ]
r 2 (cos − cos ) ∘
0
F = p B = 6 B d cos[180 −( + )] d
a
y
2 ∫ ∫ (1 + cos ) 3
1 1
(12.21)
Therefore
[ ]
2
F 2 (cos − cos ) ∘
0
= 3 d cos[180 −( + )] d (12.22)
a
Bd ∫ ∫ (1 + cos ) 3
1 1
Define the load carrying capacity coefficient C , or Sommerfeld number [18], as
p
F 2 F 2
C = = (12.23)
p
Bd 2 vB
The unit for bearing width B is selected as m. The unit of dynamic viscosity is N⋅s
−1
m −2 and the journal velocity is m s . Load carrying capacity coefficient and relative
clearance are dimensionless.