2.7 Logarithmic Decrement
49
= e
− n t 1 exp
−
2π n
p 2 − n 2
[ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
In all the above equations of displacement, p 1 =
p 2 − n 2
Substituting the values of x t 1 , x ( t 1 +T /2) and x ( t 1 +T ) from the above equations into
Eq. (a) gives
A 1
A 2
=
e
− n t 1 [ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
1 + exp
− nπ
√
p 2 −n 2
e − n t 1 [ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
exp
− 2πn
√
p 2 −n 2
+ exp
− nπ
√
p 2 −n 2
or
A 1
A 2
= exp
nπ
p 2 − n 2
or
ln
A 1
A 2
=
π n
p 2 − n 2
=
πζ
1 − ζ 2
where ζ =
n
p
and ζ being small
1 − ζ 2 = 1
Therefore,
ζ =
1
π
ln
A 1
A 2
Exercise 2
2.1 A massless beam with a concentrated mass m is shown in the figure. Determine
the natural frequency of the system.
2.2 Determine the natural frequency and the time period of the uniform cantilever
beam. The beam is considered as massless.
Prob. 2.1
49
= e
− n t 1 exp
−
2π n
p 2 − n 2
[ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
In all the above equations of displacement, p 1 =
p 2 − n 2
Substituting the values of x t 1 , x ( t 1 +T /2) and x ( t 1 +T ) from the above equations into
Eq. (a) gives
A 1
A 2
=
e
− n t 1 [ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
1 + exp
− nπ
√
p 2 −n 2
e − n t 1 [ C 1 cos p 1 t 1 + C 2 sin p 1 t 1 ]
exp
− 2πn
√
p 2 −n 2
+ exp
− nπ
√
p 2 −n 2
or
A 1
A 2
= exp
nπ
p 2 − n 2
or
ln
A 1
A 2
=
π n
p 2 − n 2
=
πζ
1 − ζ 2
where ζ =
n
p
and ζ being small
1 − ζ 2 = 1
Therefore,
ζ =
1
π
ln
A 1
A 2
Exercise 2
2.1 A massless beam with a concentrated mass m is shown in the figure. Determine
the natural frequency of the system.
2.2 Determine the natural frequency and the time period of the uniform cantilever
beam. The beam is considered as massless.
Prob. 2.1