Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 815

The general solution of this equation is

z = C 1 sin ωt + C 2 cos ωt ...(Eq. 20.7)

where C 1 and C 2 are constants.

wt

z

A sin tw

w

A

w

z, z, z. :

wA

w^2 A

t

t

wAp/2

t

p/2

w^2 A

z.

z:

(a) Vector representation

of motion

(b) Vector representation of harmonic displacement,

velocity and acceleration.

Fig. 20.3 Vector method of representing simple harmonic motion

20.2 FUNDAMENTALS OF VIBRATION

Certain fundamental aspects of Vibration essential to the study of Soil Dynamics are consid-
ered in the following subsections.

20.2.1Degree of Freedom

The ‘Degree of Freedom’ for a system is defined as the minimum number of independent
Co-ordinates required to describe the motion of the system mathematically.

A mass supported by a spring and constrained to move in only one direction is a system
with a single degree of freedom. Similarly, a simple pendulum oscillating in one plane is also
an example of a system with a single degree of freedom (Fig. 20.4).

k

M

l

q

(a) Mass supported

by a spring

(b) Simple pendulum

oscillating in one plane

Fig. 20.4 Systems with single degree of freedom