48
2 Free Vibration of Single Degree of Freedom System
Fig. 2.26 Example 2.17
δ =
1
n
ln N
(2.63)
Example 2.17 Show that the critical damping ratio ζ can be worked out from the
free vibration records, as shown in Fig. 2.26, by the expression given below
ζ =
1
π
ln
A 1
A 2
It is assumed that the amplitude a is occurring at time instant t 1 . Then, referring
to Fig. 2.26
A 1
A 2
=
a + b
b + c
=
x t 1 − x ( t 1 +T /2)
− x ( t 1 +T /2) + x ( t 1 +T )
x t 1 = e
− n t 1 [C 1 cos p 1 t 1 + C 2 sin p 1 t 1 )
(a)
x (t 1 +T /2) = exp
−n
t 1 +
π
p 2 − n 2
C 1 cos p 1
t 1 +
π
p 1
+ C 2 sin p 1
t 1 +
π
p 1
= e
− n t 1 exp
−
nπ
p 2 − n 2
[−C 1 cos p 1 t 1 − C 2 sin p 1 t 1 ]
x ( t 1 + T ) = exp
−n
t 1 +
2π
p 2 − n 2
2 Free Vibration of Single Degree of Freedom System
Fig. 2.26 Example 2.17
δ =
1
n
ln N
(2.63)
Example 2.17 Show that the critical damping ratio ζ can be worked out from the
free vibration records, as shown in Fig. 2.26, by the expression given below
ζ =
1
π
ln
A 1
A 2
It is assumed that the amplitude a is occurring at time instant t 1 . Then, referring
to Fig. 2.26
A 1
A 2
=
a + b
b + c
=
x t 1 − x ( t 1 +T /2)
− x ( t 1 +T /2) + x ( t 1 +T )
x t 1 = e
− n t 1 [C 1 cos p 1 t 1 + C 2 sin p 1 t 1 )
(a)
x (t 1 +T /2) = exp
−n
t 1 +
π
p 2 − n 2
C 1 cos p 1
t 1 +
π
p 1
+ C 2 sin p 1
t 1 +
π
p 1
= e
− n t 1 exp
−
nπ
p 2 − n 2
[−C 1 cos p 1 t 1 − C 2 sin p 1 t 1 ]
x ( t 1 + T ) = exp
−n
t 1 +
2π
p 2 − n 2