92
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.20 Time variation of
DLF
3.11.2 Response of SDF System to Gradually Applied Load
The forcing function is shown in Fig. 3.21. For 0 ≤ t ≤ t 1 , the force is linearly
varying with time, and for t > t 1 , the force remains a constant value. The solution
of this problem is to be obtained separately for these two time ranges.
(i) For t ≤ t 1
F (t) =
F 0 t
t 1
(3.79)
The equation of motion for the undamped system is
m ¨
x + kx =
F 0 t
t 1
(3.80)
The solution of the above equation is
x =
F 0
kt 1
t + A cos pt + B sin pt
(3.81)
Substituting the initial conditions at t = 0, x = 0 and ˙
x = 0 in Eq. (3.81) yields
Fig. 3.21 Gradually applied
load
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.20 Time variation of
DLF
3.11.2 Response of SDF System to Gradually Applied Load
The forcing function is shown in Fig. 3.21. For 0 ≤ t ≤ t 1 , the force is linearly
varying with time, and for t > t 1 , the force remains a constant value. The solution
of this problem is to be obtained separately for these two time ranges.
(i) For t ≤ t 1
F (t) =
F 0 t
t 1
(3.79)
The equation of motion for the undamped system is
m ¨
x + kx =
F 0 t
t 1
(3.80)
The solution of the above equation is
x =
F 0
kt 1
t + A cos pt + B sin pt
(3.81)
Substituting the initial conditions at t = 0, x = 0 and ˙
x = 0 in Eq. (3.81) yields
Fig. 3.21 Gradually applied
load