Page 77 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

74 Chapter 1 Fundamentals oF Vibration
# # #
where x p , x r , and x v are the linear velocities of the pushrod, C.G. of the rocker arm, and the valve,
#
respectively, and u r is the angular velocity of the rocker arm. # #
(i) If m eq denotes the equivalent mass placed at point A, with x eq = x, the kinetic energy of the
equivalent mass system T eq is given by
1 #
2
T eq = m eq x eq (E.2)
2
By equating T and T eq , and noting that
# # #
# # # xl 2 # xl 3 # x
x p = x, x v = , x r = , and u r =
l 1 l 1 l 1
we obtain
2 2
J r l 2 l 3
m eq = m p + + m v + m r (E.3)
l 1 2 l 1 2 l 1 2
# #
(ii) Similarly, if the equivalent mass is located at point C, x eq = x v and
1 # 1 #
2
T eq = m eq x eq = m eq x v 2 (E.4)
2 2
Equating (E.4) and (E.1) gives
2 2
J r l 1 l 3
m eq = m v + + m p ¢ ≤ + m r ¢ ≤ (E.5)
l 2 2 l 2 l 2
■

1.9 damping elements

In many practical systems, the vibrational energy is gradually converted to heat or sound.
Due to the reduction in the energy, the response, such as the displacement of the system,
gradually decreases. The mechanism by which the vibrational energy is gradually converted
into heat or sound is known as damping. Although the amount of energy converted into heat
or sound is relatively small, the consideration of damping becomes important for an accu-
rate prediction of the vibration response of a system. A damper is assumed to have neither
mass nor elasticity, and damping force exists only if there is relative velocity between the
two ends of the damper. It is difficult to determine the causes of damping in practical sys-
tems. Hence damping is modeled as one or more of the following types.

Viscous Damping. Viscous damping is the most commonly used damping mechanism
in vibration analysis. When mechanical systems vibrate in a fluid medium such as air, gas,
water, or oil, the resistance offered by the fluid to the moving body causes energy to be
dissipated. In this case, the amount of dissipated energy depends on many factors, such
as the size and shape of the vibrating body, the viscosity of the fluid, the frequency of
vibration, and the velocity of the vibrating body. In viscous damping, the damping force
is proportional to the velocity of the vibrating body. Typical examples of viscous damping