Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 817

These are known as the Principal modes of vibration of the block and are shown
schematically in Fig. 20.8.

Rocking

Lateral

Vertical

Yawing or torsional

Pitching

Y Longitudinal

X

Z

Fig. 20.8 Modes of vibration of a block
Vertical and torsional vibrations can occur independently but not others. This is be-
cause rotation about X-axis or Y-axis is always accompanied by translation along Y- or X-axis
and vice-versa, producing what is known as ‘coupled motion’.

If a combination of more than one mode of vibration occurs in a particular case, it is
referred to as ‘coupled mode’ of vibration. The analysis of such modes requires the use of
complex mathematical treatment.

20.2.3 Free Vibrations and Forced Vibrations

Bodies which have both mass and elasticity are capable of undergoing vibrations. The vibra-
tions of a body or a system may be classified as ‘Free Vibrations’ and ‘Forced Vibrations’.

‘Free Vibration’ is a vibration that occurs under the influence of forces inherent in the
system itself, without any external force. Of course, an external force or natural disturbance is
required to initiate the free vibration which continues without an external force acting con-
tinuously.

If the vibration is undamped by friction or any other forces, the body undergoes free
vibration with a frequency known as the ‘Natural frequency’ of the body or system. It is consid-
ered as the property of the body or system. Depending upon the particular mode of vibration,
the body will have a particular value of natural frequency. Thus a body or system can have as
many natural frequencies as the possible modes of vibration.

‘Forced Vibration’ is a vibration that occurs under the continuous influence of an exter-
nal force. This obviously depends upon the nature of the external force, also known as the
‘exciting force’, which may be caused by an inpulsive force or a continuous periodic force. Ham-
mer foundation produces an inpulsive force causing forced vibration of the system. A founda-
tion for a machine with rotating masses will be subjected to force a vibration caused by a
continuous periodic force.

In practice it is extremely rare that a body has free vibration at its natural frequency
undamped, since it is always subjected to some form of damping.