Geotechnical Engineering

DHARM

816 GEOTECHNICAL ENGINEERING

However, if the spring-supported mass of Fig. 20.4 (a) can also rotate in one plane its
degree of freedom is two. A two-mass two-spring system, constrained to move in one direction
without rotation, is also an example of a system with a degree of freedom of two (Fig. 20.5).

A body in space has a degree of freedom of six—three translational and three rotational
(Fig. 20.6). A flexible beam between two supports has an infinite number of degrees of freedom
(Fig. 20.7).

20.2.2 Modes of Vibration

A system with more than one degree of freedom vibrates in complex modes. However, if each
point in the system follows a definite pattern of vibration, the mode is systematic and orderly,
and is known as a ‘principal mode of vibration’. A system with a degree of a freedom of n and n
principal modes. (Of course, the number of principal modes need not always reflect the Degree
of freedom). The vibration of a block can be reduced to six modes for the purpose of analysis.
These are

Y f

q

x

Z

Y

M 1

Z 1

Z 2

M 2

Fig. 20.5 A two-mass Fig. 20.6 Body in space with six

two-spring system degrees of freedom

Fig. 20.7 A beam with infinite degree of freedom

(i) Translation along X-axis (lateral)

(ii) Translation along Y-axis (longitudinal)

(iii) Translation along Z-axis (vertical)

(iv) Rotation about X-axis (pitching)

(v) Rotation about Y-axis (rocking)

(vi) Rotation about Z-axis (yawing or Torsional)