30
2 Free Vibration of Single Degree of Freedom System
The damping constant in other cases is expressed in terms of’ a certain percentage
of critical damping. For a structure having 10% of critical damping, it will have a
damping constant c given by
c = (0.1) 2mp = 0.2 mp
(2.28)
Case III. Underdamped or Damped system (n < p)
Roots λ 1 and λ 2 are complex when n is less than p, and they are given by
λ 1, 2 = −n ± i
p 2 − n 2
(2.29)
The solution of Eq. (2.21) will be
x = A 1 e
− n+i
√
p 2 −n 2
t + A 2 e
− n−i
√
p 2 −n 2
t
(2.30)
or
x = A 1 e
− n t e
i
√
p 2 −n 2 t
+ A 2 e
− n t e
− i
√
p 2 −n 2 t
or
x = e
− n t
(A 1 + A 2 ) cos
p 2 − n 2 t + i (A 1 − A 2 ) sin
p 2 − n 2 t
or
x = e
− n t
C 1 cos
p 2 − n 2 t + C 2 sin
p 2 − n 2 t
(2.31)
where C 1 and C 2 are constants, which can be determined from initial conditions.
This system is known as underdamped or simply damped system of vibration.
If at t=0, x = x 0 and ˙
x = ˙
x 0 , then putting these conditions in Eq. (2.31), we get
x = e
− n t
x 0 cos
p 2 − n 2 t +
nx 0 + ˙
x 0
p 2 − n 2
sin
p 2 − n 2 t
(2.32)
The variation of the displacement of the damped system with time given by
Eq. (2.32) is shown in Fig. 2.14. The amplitude of vibration goes on decreasing
in an exponential manner—the reduction in magnitude for each successive cycle
gives a measure of the damping in the system. As can be seen from Eq. (2.32), the
damped angular frequency of vibration is given by
p d =
p 2 − n 2
(2.33)