Page 46 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.4 basiC ConCepts oF Vibration 43
driving, vibratory testing of materials, vibratory finishing processes, and electronic circuits
to filter out the unwanted frequencies. Vibration has been found to improve the efficiency
of certain machining, casting, forging, and welding processes. It is employed to simulate
earthquakes for geological research and also to conduct studies in the design of nuclear
reactors.

1.4 basic Concepts of Vibration

1.4.1 Any motion that repeats itself after an interval of time is called vibration or oscillation.
Vibration The swinging of a pendulum and the motion of a plucked string are typical examples of
vibration. The theory of vibration deals with the study of oscillatory motions of bodies and
the forces associated with them.

1.4.2 A vibratory system, in general, includes a means for storing potential energy (spring or
elementary parts elasticity), a means for storing kinetic energy (mass or inertia), and a means by which
of Vibrating energy is gradually lost (damper).
The vibration of a system involves the transfer of its potential energy to kinetic energy
systems and of kinetic energy to potential energy, alternately. If the system is damped, some energy
is dissipated in each cycle of vibration and must be replaced by an external source if a state
of steady vibration is to be maintained.
As an example, consider the vibration of the simple pendulum shown in Fig. 1.10.
Let the bob of mass m be released after being given an angular displacement u. At posi-
tion 1, the velocity of the bob and hence its kinetic energy is zero. But it has a poten-
tial energy of magnitude mgl11 - cos u2 with respect to the datum position 2. Since
the gravitational force mg induces a torque mgl sin u about the point O, the bob starts
swinging to the left from position 1. This gives the bob certain angular acceleration in
the clockwise direction, and by the time it reaches position 2, all of its potential energy
will be converted into kinetic energy. Hence the bob will not stop in position 2 but will
continue to swing to position 3. However, as it passes the mean position 2, a counter-
clockwise torque due to gravity starts acting on the bob and causes the bob to deceler-
ate. The velocity of the bob reduces to zero at the left extreme position. By this time,
all the kinetic energy of the bob will be converted to potential energy. Again, due to
the gravity torque, the bob continues to attain a counterclockwise velocity. Hence the
bob starts swinging back with progressively increasing velocity and passes the mean
position again. This process keeps repeating, and the pendulum will have oscillatory
motion. However, in practice, the magnitude of oscillation 1u2 gradually decreases and
the pendulum ultimately stops due to the resistance (damping) offered by the surround-
ing medium (air). This means that some energy is dissipated in each cycle of vibration
due to damping by the air.