248
6 Free Vibration of Multiple Degrees of Freedom System
Fig. 6.25 Free interface
nodes
Once Eq. (6.182) is solved, the displacements of each substructure can then be
found from Eq. (6.1). The method is very elegant and possesses good convergence
characteristic with the increase of number of component modes.
6.21.2 Free Interface Method
In the free interface method, the following transformation is used for reducing the
number of degrees of freedom in a substructure
{x}
S
= [] N {ξ N }
(6.184)
[] N is obtained from the solution of the eigenvalue problem of the substructure
[K ]
S
− p
2
[M]
S
{φ } = {0}
(6.185)
Free interface modes of substructure 1 of Fig. 6.23 are shown in Fig. 6.25. [φ] N
are modes of the substructure with interface boundary.
Substructures having both ends free will result in a rigid body mode which is to
be treated as a mode with zero frequency.
Substituting Eq. (6.184) into Eq. (6.162) yields
T S =
1
2
{ ˙
ξ N }
S
T [M]
S
{ ˙
ξ N }
S
(6.186)
U S =
1
2
{ξ N }
S
T [K ]
S
{ξ N }
S
(6.187)
where
[M]
S
= [φ ]
T
N [M]
S
[φ ] N
(6.188)
and