Civil Insight: A Technical Magazine Volume 3 | Page 32

Bhatt M.R.
Civil Insight (2019) 29-36
3) Results and Discussions
For the assumed SDOF system in Fig. 1, responses of undamped and damped systems were plotted and
discussed for different frequency ratio, which are shown in the figures below. Further, considering different
values of the critical damping ratio, dynamic response factors were plotted and the results were obtained.
3.1) Response of undamped system at ࣓ = ࣓ ࢔
Considering the fundamental natural period T n is 2 seconds, the natural and forcing frequency will be equal
to ߱ = ߱ ௡ = 3.142 rad/sec.
Response at ߱ = ߱ n
60
40
20
0
-20
0
5
10
15
20
25
30
-40
Time (sec)
-60
Fig. 2. Response of undamped system to sinusoidal force of frequency ߱ = ߱ ௡ , with initial displacement
and velocity as zero
100
80
60
40
20
0
-20 0
-40
-60
-80
-100
Response at ߱ = ߱ n
10
20
30
40
50
60
70
Time (sec)
Fig. 3. Response of undamped system to sinusoidal force of frequency ߱ = ߱ ௡ , with initial displacement
and velocity as zero (for longer time)
The Fig. 2 and Fig. 3 above show the responses of undamped resonant frequency case for different time
periods; it can be seen that the amplitude of the vibration is continously increasing with increase in time.
Deformation amplitude grows indefinitely; it becomes infinite after a considerably long time. This is an
academic result, but for the real structures, with the increasing amplitude, at some point of time, the system
will fail either by brittle failure or by ductile yielding failure (Chopra, 2007).
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