Page 79 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

76 Chapter 1 Fundamentals oF Vibration
damping Constant of parallel plates separated by Viscous Fluid
example 1.13
Consider two parallel plates separated by a distance h, with a fluid of viscosity m between the plates.
Derive an expression for the damping constant when one plate moves with a velocity v relative to the
other as shown in Fig. 1.41.

Solution: Let one plate be fixed and let the other plate be moved with a velocity v in its own plane.
The fluid layers in contact with the moving plate move with a velocity v, while those in contact with
the fixed plate do not move. The velocities of intermediate fluid layers are assumed to vary linearly
between 0 and v, as shown in Fig. 1.41. According to Newton’s law of viscous flow, the shear stress
1t2 developed in the fluid layer at a distance y from the fixed plate is given by
du
t = m (E.1)
dy
where du>dy = v>h is the velocity gradient. The shear or resisting force (F) developed at the bottom
surface of the moving plate is

mAv
F = tA = (E.2)
h
where A is the surface area of the moving plate. By expressing F as

F = cv (E.3)
the damping constant c can be found as
mA
c = (E.4)
h

Surface area of plate A
dx
v
dt
Viscous vy F (damping force)
h fluid u h
y

FiGure 1.41 Parallel plates with a viscous fluid in between.

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Clearance in a bearing
example 1.14
A bearing, which can be approximated as two flat plates separated by a thin film of lubricant
(Fig. 1.42), is found to offer a resistance of 400 N when SAE 30 oil is used as the lubricant and
2
the relative velocity between the plates is 10 m/s. If the area of the plates (A) is 0.1 m , determine the
clearance between the plates. Assume the absolute viscosity of SAE 30 oil as 50 mreyn or 0.3445 Pa-s.