Page 57 - Dynamic Loading and Design of Structures

Page 40

2.2.2.1 Free vibrations
The equation of motion for an SDOF system in the presence of damping is

(2.27)

and its solution without external loading (F(t)=0) is given below as

(2.28)

We also define the coefficient of damping and the damped natural frequency as follows:

(2.29)

(2.30)

,
There are three possibilities for βnamely

(2.31)

which correspond to underdamped, critically damped and overdamped conditions. If
and eqn (2.28) becomes

(2.32)

The displacement is no longer a periodic function of time and the oscillator simply returns to
its original position without executing any vibrations. From the condition

(2.33)

we may compute the coefficient of critical damping as

(2.34)

Following that, the damping ratio is defined