DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 821

A =

P

M

o

n

n

ω ω

ω

2 2

1 − 2

F

HG

I

KJ

...(Eq. 20.20)

But

P

M

o

ωn

2 = Ast ...(Eq. 20.21)

where Ast = deflection of the system under Po, applied statically. The ratio

ω

ωn

F

HG

I

KJ is called the

frequency ratio, ξ.

∴ A = Ast

()1−ξ

...(Eq. 20.22)

The factor^1

() 1 −ξ^2

is called the ‘magnification factor’, ηo. It is the ratio of the dynamic

amplitude to the static displacement. A plot between ξ and ηo is shown in Fig. 20.11.

4

3

2

1

0

h^0

Zone of

resonance

Ö (^2234)
d=0,h 0 =1
d 1, h ¥
d Ö2, h
d ¥, h

==1
==0
0
0
0
h 0
Fig. 20.11 Frequency ratio vs magnification factor
When the exciting frequency approaches the natural frequency of the system (ξ = 1), the
magnification factor, and hence the amplitude of vibration tend to become infinite, leading to
resonance. If the frequency ratio is more than 1, there will be steep decrease of the magnifica-
tion factor.
It is obvious that resonant conditions should be avoided.
20.2.8 Free Vibrations with Damping
Assuming that in a system undergoing free vibrations viscous damping is present, a “Mass-
spring-Dashpot” system can serve as the relevant mathematical mode for analysis (Fig. 20.12).
The ‘dashpot’ is the simplest mathematical element to simulate a viscous damper. The force in
the dashpot under dynamic loading is directly proportional to the velocity of the oscillating
mass.
The equation of motion is
Mz c z kz&&++.& = 0 ...(Eq. 20.23)
where c represents the coefficient of viscous damping expressed as force per unit velocity.