2.4 Free Damped Vibration of Sdf System
29
Fig. 2.12 Overdamped
motion
only for a short time, which however goes on decreasing in an exponential manner
with time. In this case, the viscous resistance of the body is so large that when
it is displaced from the equilibrium position, it only creeps back to that position.
The displacement finally vanishes as t approaches infinity. This system is known as
overdamped system.
Case II. Critically damped system (n = p)
When n = p, roots λ 1 and λ 2 of Eq. (2.24) are real, negative and equal, that is,
repeating.
The solution of Eq. (2.21) is
x = (A 1 + A 2 t ) e
λ t
(2.26)
where λ 1 = λ 2 = λ.
The displacement of the mass reaches zero asymtotically with time for this case,
as shown in Fig. 2.13. The damping in the system is great enough not to set up
oscillatory motion. The system is known as critically damped system.
At critical damping, damping coefficient c is given by
c = 2mp
(2.27)
Fig. 2.13 Critically damped
motion
29
Fig. 2.12 Overdamped
motion
only for a short time, which however goes on decreasing in an exponential manner
with time. In this case, the viscous resistance of the body is so large that when
it is displaced from the equilibrium position, it only creeps back to that position.
The displacement finally vanishes as t approaches infinity. This system is known as
overdamped system.
Case II. Critically damped system (n = p)
When n = p, roots λ 1 and λ 2 of Eq. (2.24) are real, negative and equal, that is,
repeating.
The solution of Eq. (2.21) is
x = (A 1 + A 2 t ) e
λ t
(2.26)
where λ 1 = λ 2 = λ.
The displacement of the mass reaches zero asymtotically with time for this case,
as shown in Fig. 2.13. The damping in the system is great enough not to set up
oscillatory motion. The system is known as critically damped system.
At critical damping, damping coefficient c is given by
c = 2mp
(2.27)
Fig. 2.13 Critically damped
motion