Page 76 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.8 mass or inertia elements 73
we obtain the equivalent mass of the system as
2 2 2 2
J p 1 m 1 l 1 m 2 l 1 1 m c l 1 l 1
m eq = m + + + + + m c 2 (E.4)
r p 2 3 r p 2 r p 2 2 r p 2 r p
■

Cam-Follower mechanism
example 1.12
A cam-follower mechanism (Fig. 1.39) is used to convert the rotary motion of a shaft into the oscil-
lating or reciprocating motion of a valve. The follower system consists of a pushrod of mass m p , a
rocker arm of mass m r , and mass moment of inertia J r about its C.G., a valve of mass m v , and a valve
spring of negligible mass [1.28–1.30]. Find the equivalent mass 1m eq 2 of this cam-follower system
by assuming the location of m eq as (i) point A and (ii) point C.

Solution: The equivalent mass of the cam-follower system can be determined using the equiva-
lence of the kinetic energies of the two systems. Due to a vertical displacement x of the pushrod,
the rocker arm rotates by an angle u r = x>l 1 about the pivot point, the valve moves downward by
x v = u r l 2 = xl 2 >l 1 , and the C.G. of the rocker arm moves downward by x r = u r l 3 = xl 3 >l 1 . The
kinetic energy of the system (T) can be expressed as 2

1 # 1 # 1 # 1 #
2
2
2
T = m p x p + m v x v + J r u r + m r x r 2 (E.1)
2 2 2 2
l 3
x x p Rocker arm
u r (mass moment of inertia, J r )
B
O G
A
l 1 l 2
Pushrod
(mass m ) x v
p
Valve
spring
Valve
Roller (mass m )
v
follower

Cam
C
Shaft
FiGure 1.39 Cam-follower system.

1
2 If the valve spring has a mass m s , then its equivalent mass will be m s (see Example 2.8). Thus the kinetic energy
#
3
2
1 1
of the valve spring will be 1 m s 2x v .
2 3