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2 Free Vibration of Single Degree of Freedom System
2.4 Free Damped Vibration of Sdf System
From Eq. (2.1), it follows that the equation of motion for free vibration for a damped
SDF system is
m ¨
x + c ˙
x + kx = 0
(2.21)
Equation (2.21) is a linear differential equation of second order and can be solved
by using standard procedure.
The general solution of the equation can be assumed in the following form
x = Ae
λ t
(2.22)
Substituting x and its time derivatives from Eq. (2.22) into Eq. (2.21) and
rearranging the terms, we get
Aλ
2 e
λ t
+
c
m
Aλe
λ t
+
k
m
Ae
λ t
= 0
(2.23)
or
λ
2
+ 2nλ + p
2
= 0
where
n =
c
2m
and p
2
=
k
m
Equation (2.23) is a quadratic equation, and its solution is given by
λ 1, 2 = −n ±
n 2 − p 2
(2.24)
Relative values of n and p will govern the resulting solution. Three cases may
arise, and they are discussed below one after another.
Case I. Overdamped system (n > p)
Roots λ 1 and λ 2 of Eq. (2.24) are real and negative when n is greater than p. The
solution of the equation is
x = A 1 e
λ 1 t
+ A 2 e
λ 2 t
(2.25)
The solution is shown graphically in Fig. 2.12. The solution does not con-tain any
periodicity; as such it does not represent a vibratory motion. The roots of Eq. (2.24)
are unequal and two terms on the right-hand side represent motions which decay
exponentially at two different rates. The motion associated with root λ 1 predominates