12.9 Periodic Structures
485
( [W ]
H
[M
e
] [W ] )
H
= [W ]
H
[M
e
] [W ]
(12.193)
(Note, [A] is Hermitian if [A]
H
= [A])
Relationship given by Eq. (12.189) can be derived in another way.
Equation (12.180) can be written as
(12.194)
In Eq. (12.194), [K
e
] and [M
e
] matrices are partitioned corresponding to the
degrees of freedom of three types of modes—those on the left hand boundary, all
interior nodes and those on the right hand boundary.
Three matrices equations in terms of partitioned matrices can be written as
[K
e
L L ] {δ
e
L } + [K
e
L I ] {δ
e
I } + [K
e
L R ] e
μ
{δ
e
L } − p
2
( [M
e
L L ] {δ
e
L }
+ [M
e
L I ] {δ
e
I } + [M
e
L R ] e
μ
{δ
e
L } ) = {F
e
L }
(12.195)
( [K
e
I L ] {δ
e
L } + [K
e
I I ] {δ
e
I } ) + [K
e
I R ] e
μ
{δ
e
L } − p
2
( [M
e
I L ] {δ
e
L }
+ [M
e
I I ] {δ
e
I } + [M
e
I R ] e
μ
{δ
e
L } ) = {0}
(12.196)
( [K
e
RL ] {δ
e
L } + [K
e
R I ] {δ
e
I } + [K
e
R R ] e
μ
{δ
e
L } ) − p
2
([M
e
RL ] {δ
e
L }
+ [M
e
R I } {δ
e
I } + [M
e
R R ] e
μ
{δ
e
L } ) = − e
μ
{F
e
L }
(12.197)
Eliminating {F
e
L } from Eqs. (12.195) and (12.197), we get
( [K
e
L L ] + [K
e
R R ] + [K
e
I R ] e
μ
+ e
− μ
[K
e
R I ]) {δ
e
L } + ( [K
e
L I ] + e
− μ
[K
e
R I ] )
{δ
e
I } + ( [M
e
L L ] + [M
e
R R ] + e
μ
[M
e
L R ] + e
− μ
[M
e
R I ] ) {δ
e
L }
+ ( [M
e
L I ] + e
− μ
[M R I ] ) {δ
e
I } = {0}
(12.198)
Writing Eqs. (12.196) and (12.198) in partitioned forms, we get