Geotechnical Engineering

DHARM

826 GEOTECHNICAL ENGINEERING

To determine the conditions corresponding to maximum amplitude, Eq. 20.48 may be
differentiated with respect to ξ, equated to zero, and solved for ξ. One obtains

ξ = 12 − D^2 ...(Eq. 20.49)
It is clear from this equation that if D decreases, ξ increased and vice versa. Resonance
condition is said to occur when the peak amplitude occurs. Hence, the magnification factor at
resonance, η1 max is got by substituting the value of ξ from Eq. 20.49 in Eq. 20.48.

∴ η1 max =

1

21 DD−^2

...(Eq. 20.50)

From this equation, it can be seen that the larger the damping ratio, the smaller the
magnification factor at resonance, and vice versa.

The relationship between ξ and η 1 (or A) for varying D is shown in Fig. 20.15.

5 4 3 2 1 0

h 1

12345

x

D=0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 20.15 Magnification factor versus frequency ratio
It can be observed that the maximum value of η 1 , and hence the peak amplitude occurs
at a value of ξ less than unity when damping is present. As the Damping ratio, D, increases the
value of ξ for peak amplitude deviates more from unity. The corresponding frequency at which
peak amplitude occurs at a certain value of damping is known as the resonant frequency for
the damped case.

It may be recalled that, without damping, the peak amplitude which occurs when ξ = 1,
is infinite. The effect of damping is to make the peak amplitude finite and make the frequency
ratio for peak amplitude deviate from unity. In other words, what is called the resonant fre-
quency is different in the undamped and damped cases.

Quadratic Excitation

In this type of excitation, the exciting force is proportional to the square of the frequency. This
is caused by the rotation of unbalanced masses (Fig. 20.16) in an oscillator.