Page 83 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

70 Harmonically Excited Vibration Chap. 3

(a) By substituting into Eq. (3.1-5), the amplitude of vibration is
350
700 X 10
50

1 - 2 X 0.20 X
\ 13.32 j 13.32
= 3.79 X 10~^ m
= 0.0379 mm
(b) The transmissibility from Eq. (3.6-2) is

r * (2 X 0.20 X - ,1 ^ ) “
TR = = 0.137

/ [ ' - (t o î )’] +(2x»-2»XTÎ32f
(c) The transmitted force is the disturbing force multiplied by the transmissibility.
= 350 X 0.137 = 47.89 N

3.7 ENERGY DISSIPATED BY DAMPING

Damping is present in all oscillatory systems. Its effect is to remove energy from
the system. Energy in a vibrating system is either dissipated into heat or radiated
away. Dissipation of energy into heat can be experienced simply by bending a piece
of metal back and forth a number of times. We are all aware of the sound that is
radiated from an object given a sharp blow. When a buoy is made to bob up and
down in the water, waves radiate out and away from it, thereby resulting in its loss
of energy.
In vibration analysis, we are generally concerned with damping in terms of
system response. The loss of ener^^y from the oscillatory system results in the decay
of amplitude of free vibration. In steady-state forced vibration, the loss of energy is
balanced by the energy that is supplied by the excitation.
A vibrating system can encounter many different types of damping forces,
from internal molecular friction to sliding friction and fluid resistance. Generally,
their mathematical description is quite complicated and not suitable for vibration
analysis. Thus, simplified damping models have been developed that in many cases
are found to be adequate in evaluating the system response. For example, we have
already used the viscous damping model, designated by the dashpot, which leads to
manageable mathematical solutions.
Energy dissipation is usually determined under conditions of cyclic oscilla­
tions. Depending on the type of damping present, the force-displacement relation­
ship when plotted can differ greatly. In all cases, however, the force-displacement
curve will enclose an area, referred to as the hysteresis loop, that is proportional to