246
6 Free Vibration of Multiple Degrees of Freedom System
The energy expression for each substitution is given by
T S =
1
2
({ ˙
x}
S
)
T
[M]
S
{ ˙
x}
S
(6.163)
U S =
1
2
({x}
S
)
T
[K ]
S
{x}
S
(6.164)
where superscript S denotes a substructure.
Using transformation given by Eq. (6.158) in Eqs. (6.163) and (6.164) yields
T S =
1
2
{ ˙
ξ }
S
T [T F ]
S
T [M]
S
[T F ]
S
{ ˙
ξ }
S
(6.165)
U S =
1
2
{ ˙
ξ }
S
T [T F ]
S
T [K ]
S
[T F ]
S
{ξ }
S
(6.166)
or,
T S =
1
2
{ ˙
ξ }
S
T [M]
S
{ ˙
ξ }
S
(6.167)
U S =
1
2
{ξ)
S
T [K ]
S
{ξ }
S
(6.168)
where
[M]
S
=
[T F ]
S
T [M]
S
[T F ]
S
(6.169)
[K ]
S
=
[T F ]
S
T [K ]
S
[T F ]
S
(6.170)
[M]
S and [K ]
S can be reduced to the following form with the help of Eq. (6.158).
[M]
S
=
[M] N N [M] N B
[M] B N [M] B B
(6.171)
and
[K ]
S
=
[K ] N N [0]
[0] [K ] B B
(6.172)
Now,
[M] N N = []
T
N [M] I I [] N
(6.173)