The torque due to the frictional resistance of the oil film between a rotating shaft and its bearing is found to be dependent on the force F normal to the shaft, the speed of rotation N of the shaft, the dynamic viscosity ? of the oil, and the shaft diameter D. Establish a correlation among the variables by using dimensional analysis.

Question

GIVEN
The oil film between a rotating shaft and its bearing
The torque (T) due to frictional resistance is a function of normal force (F), speed of rotation (N),
dynamic viscosity (?), and shaft diameter (D)
FIND
A correlation among the variables
ASSUMPTIONS
Steady state

Answer 1

The Buckingham ? Theorem can be used to find the correlation. The

primary dimensions of the variables are listed below



There are 5 variables and 3 primary dimensions. Therefore, two dimensionless groups are needed to

correlate the variables



In terms of the primary dimensions



Equation the sum of the exponents of each primary dimension to zero

For ?: a + b + d = 0 [1]

For L: 2a + b – d + e = 0 [2]

For t: 2a + 2b + c + d = 0 [3]

By inspection of equation [1] and [3]: c = d

There are five unknowns but only 3 equations. Therefore, the value of two of the exponents can be

chosen for each dimensionless group.

For ?1: Let a = 1 and b = 0

From equation [1] d = – 1 = c

From equation [2] e = – 3



For ?2: Let a = 0 and b = 1

From equation [1] d = – 1 = c

From equation [2] e = – 2



From Equation (5.24)