Geotechnical Engineering

DHARM

820 GEOTECHNICAL ENGINEERING

ωn in this case is called ‘Natural Circular Frequency’ of the system.

ωn = kM/ rad/s ...(Eq. 20.13)

and fn =

ω

ππ

n kM

2

1

2

= / ...(Eq. 20.14)

The period, Tn =

2 π 2

ω

π

n

= Mk/ ...(Eq. 20.15)

The time-displacement curve, which is known as the response curve of the system is
shown in Fig. 20.9 (c). Free vibrations may be initiated by either an initial displacement or an
initial velocity (due to impact). The final solution depends upon these initial conditions.

20.2.7 Forced Vibrations without Damping

If a mass supported by a spring is subjected to an exciting force, the
system undergoes forced vibrations. Such an exciting force may be
caused by unbalanced rotating machinery or by other means.

In the analysis that follows, it is assumed that the exciting
force is periodic and that it may be expressed as

P = Po sin ωt ...(Eq. 20.16)

where Po is the maximum value of the exciting force and ω is the
circular frequency of the exciting force in rad/s. The system is shown
in Fig. 20.10.

The equation of motion for the system may be written as

Mz kz.&&+. = Po sin ωt ...(Eq. 20.17)

or &&zzP sin
M

+=ωωn^2 o t ...(Eq. 20.18)

since

k

M

= ωn^2
The solution of Eq. 20.18 includes the solution for free vibrations (Eq. 20.12), along with
the solution which satisfies the right hand side of Eq. 20.18. The solution may be obtained by
parts as the sum of the complementary function and the particular integral. The complementary
function which represents the free vibration does not exist in this situation and the particular
integral alone is of interest.

Since the applied force is harmonic, the motion of the system may be taken to be har-
monic. Thus the particular integral may be taken as

z = A sin ωt ...(Eq. 20.1)

By substituting this in Eq. 20.18, we may show that

A =

P

M

o

()ωωn

(^22) − ...(Eq. 20.19)
It follows that the frequency of a forced vibration is equal to that of the exciting force.
(This is the same as the speed of machine, in case it is a machine that is being dealt with).
Equation 20.19 may be rewritten as
M
P sin to w
K
Fig. 20.10 Forced
vibration—undamped
mass-spring system