D33 = 3232 + p(333 + 3332 )+ p3333 = 44 + 33
Therefore, Eq. A12 is written as
c,11 +p2 55 P(C46 +C25) C46 +P2 C35 rV1
22 +p2 44 p(23 44) 2 = 0 (A13)
Sym c44 +p2c33- V3
c55 0 c35 0 c46 +C 25 0 C 11 0 46 V1
S0 44 0 + PL 46 +25 0 c23c44 + 0 22 0 v2 =0(A14)
c 35 0 C33 J 0 C23 + C44 0 c -46 0 C44 V3
The constant PL exists in every term of the coefficient matrix in Eq. A14. Although
it is possible solve for the root of the characteristic equation, numerical solutions to this
type of problem yield more accurate eigenvalues. Hence Eq. A14 is further developed
into an eigenvalue problem where PL are complex eigenvalues with corresponding
complex eigenvectors, vj where the L subscript associates the eigenvector with the
appropriate eigenvalue (L=1,2,3). The equation is simplified as
[p2[A'] + p[B'] + [C']][v]= 0 (A15)
Multiply equation by the inverse of the coefficient matrix [A ].
[p2[I]+ p[B]+[C]][v] 0 (A16)
The eigenvalue equation is
[[A]6-6 A[I]][Z]= 0 (A17)
where the array [A] can be constructed such that
A LI-[B] -[C]
[A]= [B] [C] (A18)
I ] [0] ]66