Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 819

(v) ‘Radiation’, ‘dispersion’ or ‘geometric’ damping. In the case of machine foundation
resting on soil, damping occurs due to the loss of energy on two counts. First, some energy loss
occurs by the absorption of energy into the system, reflected by the hysterisis in the stress-
strain relationship; damping caused by this internal loss of energy is called ‘internal damping’,
already given in (iii). Next, the dissipation of energy by wave propagation, radiating away into
the soil mass, causes damping effect. This is known as ‘radiation’, ‘dispersion’, or ‘geometric’
damping.

Negative Damping
Generally speaking, damping is positive, so that energy is always absorbed from the system by
damping devices. If the system draws energy from some source or is supplied energy, the
amplitude continues to increase, leading to instability. Such a system is said to be negatively
damped. The build-up of amplitudes of transmission line wires, or tall poles or suspension
bridges under the action of uniform wind flow at critical speeds are examples of negatively
damped systems. In structural systems subjected to dynamic forces due to an earthquake or a
blast, the damping is always positive.

20.2.6 Free Vibrations without Damping

The mathematical model consists of a mass supported by a weightless spring (Fig. 20.9) with
single degree freedom.

M z M z

T

O

t

(a) Equilibrium position (b) Displaced position (c) Response curve
Fig. 20.9 Free vibrations—undamped-mass spring system
If z is the vertical displacement of the system from its equilibrium position, and k is the
spring constant, applying Newton’s law of motion, the equation of motion is

Mz kz.&&+. = 0 ...(Eq. 20.9)

or &&z k
M

+F z

HG

I

KJ

= 0

or &&zz+ωn^2 = 0 ...(Eq. 20.10)

where ωn^2 = k/M ...(Eq. 20.11)

Eq. 20.10 is a homogeneous linear differential equation and the solution is given by

z = C 1 sin ωnt + C 2 cos ωnt ...(Eq. 20.12)

where C 1 and C 2 are constants which can be evaluated from the initial conditions of the system.

The equation also represents simple harmonic motion expressed by Eq. 20.7, ωn being
the circular frequency. Therefore, the free vibration of a mass resting on a spring and sub-
jected to inertial forces only can be represented by a simple harmonic motion.