7.5 Damping in MDF Systems
275
[C] =
1.035 −0.335
−0.335 0.335
[]
T [C][] =
1.000 2.414
1.000 −0.414
1.035 −0.335
−0.335 0.335
1.000 1.000
2.414 −0.414
=
1.370 0
0 1.370
{F(t)} =
P cos 4π t
0
[]{F(t)} =
1.000 2.414
1.000 −0.414
P cos 4π t
0
= P cos 4π t
1
1
Therefore, Eq. (7.27) for this problem for r = 1 is as follows:
6.827 ¨
ξ 1 + 1.37 ˙
ξ 1 + 292.75 × 6.827ξ 1 = P cos 4π t
(a)
Before, we proceed further, let us check the damping ratio for this mode
ζ =
1.37
2 × 6.827 × 17.11
= 0.00586
Equation (7.27) is written after dividing both sides by 6.827 as
¨
ξ 1 + 0.2 ˙
ξ 1 + 292.75ξ 1 =
P
6.827
cos 4π t
(b)
Therefore, for the steady-state vibration
ξ 1 =
P
6.827
292.75 − (4π )
2
2 + (2 × 0.1 × 4π )
2
cos 4π t
=
P
1737.67
cos 4π
Similarly
1.171 ¨
ξ 2 + 1.37 ˙
ξ 2 + 1707.34 × 1.171 ξ 2 = P cos 4π t
or