Geotechnical Engineering

DHARM

814 GEOTECHNICAL ENGINEERING

A graphical representation of the simple harmonic motion of the weight is shown in

Fig. 20.2. The actual line of oscillation of the point p in the vertical direction may be taken as

the projection on the vertical diameter of the point ‘a’ rotating at uniform angular velocity

about the circle with the centre at O (Fig. 20.2 (a)). The displacement versus time is shown in

(Fig. 20.2 (b)).

p

Displacement

p

a

O

wt

–zmax

+zmax

Displacement z

wpt= /2

p 3/2p 2 p

Cycle t = 2wp

T=2 /pw

Time t

O

(a) Circulation motion (b) Displacement versus time
Fig. 20.2 Graphical representation of simple harmonic motion
The equation of motion is represented by a sine function
z = A sin ωt ...(Eq. 20.1)
where ω is the circular frequency in radians per unit time. This is also the angular velocity of
point ‘a’ around 0 in Fig. (20.2 (a)).
The cycle of motion is completed when ωt = 2π.
Therefore,

the period, T =

2 π

ω

...(Eq. 20.2)

The number of cycles per unit of time, or the frequency, is

f =

1

T 2

= ω
π ...(Eq. 20.3)
The number of cycles per second is called ‘Hertz’ (Hz). Successive differentiation of Eq.
20.1 gives

dz

dt

=z& = ωA cos ωt = ωω

π

AtsinFHG + IKJ

2

...(Eq. 20.4)

and

dz

dt

z

2

2 =&& = – ω

(^2) A sin ωt = ω (^2) A sin (ωt + π) ...(Eq. 20.5)
It is obvious that velocity leads the displacement by 90° and acceleration leads the dis-
placement by 180°.
If a vector of length A is rotated counterlockwise about the Origin as shown in Fig. 20.3
(a) its projection on to the vertical axis would be equal to A sin ωt which is exactly the expres-
sion for displacement given by Eq. 20.1. Similarly it can be easily understood that the velocity
can be represented by the vertical projection of a vector of length ωA positioned 90° ahead of
displacement vector, and acceleration by a vector of length ω^2 A located 180° ahead of the
displacement vector. A plot of all these three is shown in Fig. 20.3 (b). The differential Equa-
tion of motion is &&z+ω^2 A sin ωt = 0 or
&&z+ω^2 z = 0 ...(Eq. 20.6)