Page 82 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.9 dampinG elements 79
From the definition of the rotational damping constant of the bearing 1c t 2:
T
c t = (E.5)
v
we obtain the desired expression for the rotational damping constant as
3
2pmR l
c t = (E.6)
d
Note: Equation (E.4) is called Petroff’s law and was published originally in 1883. This equation is
widely used in the design of journal bearings [1.43].

■

piston-Cylinder dashpot
example 1.16
Develop an expression for the damping constant of the dashpot shown in Fig. 1.44(a).
Solution: The damping constant of the dashpot can be determined using the shear-stress equation
for viscous fluid flow and the rate-of-fluid-flow equation. As shown in Fig. 1.44(a), the dashpot
consists of a piston of diameter D and length l, moving with velocity v 0 in a cylinder filled with a
liquid of viscosity m [1.24, 1.32]. Let the clearance between the piston and the cylinder wall be d. At
a distance y from the moving surface, let the velocity and shear stress be v and t, and at a distance
1y + dy2, let the velocity and shear stress be 1v - dv2 and 1t + dt2, respectively (see Fig. 1.44(b)).
The negative sign for dv shows that the velocity decreases as we move toward the cylinder wall. The
viscous force on this annular ring is equal to
dt
F = pDl dt = pDl dy (E.1)
dy

P P

Cylinder Cylinder
y y
dy dy

l v 0 Piston l v 0 Piston
d D d d D d
Viscous Viscous
fluid fluid
(a) (b)

FiGure 1.44 A dashpot.