3.17 Support Motion
111
3.17.1 Displacement Approach
Consider the frame of Fig. 3.36 subjected to support motion x s . The beam of the
frame is considered to be rigid and having total mass m. The mass of the column is
negligible. The system is assumed to be undamped. The equation of motion is
m ¨
x + k (x − x s ) = 0
(3.134)
or
m ¨
x + kx = kx s
(3.135)
Before proceeding further, Eq. (3.91) is modified. Let F (τ ) = F 1 f (τ ), where
F 1 is the amplitude of the force and f (τ ) time function. Noting that x st = F 1 /k,
Eq. (3.91) is rewritten as
x = x st p
t
0
f (τ ) sin p (t − τ ) dτ
(3.136)
Equation (3.135) is of the following form
m ¨
x + kx = F(t)
(3.137)
If we assume that the support displacement is sinusoidally varying with time, then
x s = x s0 sin ωt, and solution of Eq. (3.135) can be obtained from Eq. (3.136)
x = x s0 p
t
0
sin ωτ sin p (t − τ ) dτ
(3.138)
or
Fig. 3.36 A portal frame
111
3.17.1 Displacement Approach
Consider the frame of Fig. 3.36 subjected to support motion x s . The beam of the
frame is considered to be rigid and having total mass m. The mass of the column is
negligible. The system is assumed to be undamped. The equation of motion is
m ¨
x + k (x − x s ) = 0
(3.134)
or
m ¨
x + kx = kx s
(3.135)
Before proceeding further, Eq. (3.91) is modified. Let F (τ ) = F 1 f (τ ), where
F 1 is the amplitude of the force and f (τ ) time function. Noting that x st = F 1 /k,
Eq. (3.91) is rewritten as
x = x st p
t
0
f (τ ) sin p (t − τ ) dτ
(3.136)
Equation (3.135) is of the following form
m ¨
x + kx = F(t)
(3.137)
If we assume that the support displacement is sinusoidally varying with time, then
x s = x s0 sin ωt, and solution of Eq. (3.135) can be obtained from Eq. (3.136)
x = x s0 p
t
0
sin ωτ sin p (t − τ ) dτ
(3.138)
or
Fig. 3.36 A portal frame