Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 827

e

w

e

w

w w

Me 1 w^2 Me 1 w^2

M 1 M 1

(a) Rotation of unbalanced masses (b) Counteracting forces
Fig. 20.16 Quadratic excitation due to rotation of unbalanced masses
The exciting moment, Me.e, may be varied by varying either the total unbalanced mass
Me or the eccentricity e. The periodic force is not constant unlike the previous case.

The rotating force of each mass is M 1 eω^2. The total force in the vertical position is 2M 1 eω^2
or Meeω^2 where Me is the total unbalanced mass (equal to 2 M 1 ). The vibrating force at any
position may be represented by

P = Meeω^2 sin ωt = Ptosinω ...(Eq. 20.51)

where Po = Meew^2 ...(Eq. 20.52)

The periodic force is expressed by Eq. 20.16, replacing Po by Po for a frequency-depend-

ent exciting force.

∴ P = Posin ωt ...(Eq. 20.53)

We may write

P

k

Me

k

Me

M

M

k

Me

M

Me

M

oe e e

n

e

==F

HG

I

KJ

F

HG

I

KJ

=

F

HG

I

KJ

=

U

V

||

W

|

|

ω

ω

ω

ω

ξ

2 2 2

.^2 ...(Eq. 20.54)

The differential equation of motion and its solution are the same as those in the previ-
ous case in as much as these are independent of the method of applying the exciting force.

The amplitude may be got as follows, using Eq. 20.46, substituting Po for Po, and using

Eq. 20.54, and simplifying further:

A =

Me

M D

e.

()

ξ

ξξ

2

14 −+^22 2 2 ...(Eq. 20.55)
Analysing in the same manner as in the previous case, the maximum amplitude occurs
when

ξ =

1

12 − D^2

...(Eq. 20.56)

From this it can be seen that as D decreases, ξ decreases and vice versa. Defining the
magnification factor, η 2 , as