7.6 Response of MDF Systems to Support Motion
285
The dynamic load factor is determined as follows:
(DLF) a,r = p r
t
o
f (τ ) sin p r (t − τ )dτ
= p r
t
o
sin ωτ sin p r (t − τ )dτ
= p
2
r
sin ωt
p 2
r − ω 2 −
ω
p r
sin p r t
p 2
r − ω 2
=
1
1 −
ω 2
p 2
r
sin ωt −
ω
p r
sin p r t
For the first normal mode, when r = 1
(DLF) a,1 =
1
1 −
(4π) 2
9 2
sin 4π t −
4π
9
sin 9t
= −1.053(sin 4π t − 0.535 sin 23.5t)
Similarly,
(DLF) a,2 = 1.400(sin 4π t − 0.535 sin 23.5t)
Therefore,
z 1 = φ
(1)
1 B 1
¨
x so
p
2
1
(DLF) a,1 + φ
(2)
1 B 2
¨
x so
p
2
2
(DLF) a,2
= 1 × 0.783 ×
3.175
81
(DLF) a,1 + 1.00 × 0.216 ×
3.175
23.5 2 (DLF) a,2
= 0.307(DLF) a,1 + 0.00124(DLF) a,2
= 0.0307[−1.053(sin 4π t − 1.396 sin 9t)]
+ 0.00124[1.40(sin 4π t − 0.535 sin 23.5t)]
= −0.0306 sin 4π t − 0.0428 sin 9t − 0.00066 sin 23.5t
Similarly,
z 2 = (1.57)(0.783)
3.175
81
(DLF) a,1 + (−1.06)(0.216)
3.175
23.5 2 (DLF) a,2
= 0.0482(DLF) a,1 − 0.00131(DLF) a,2