Page 198 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.6 Orthogonality of Eigenvectors 185

to determine the second term, which is
6
FI
^12^22*^21 = — T 0 2

TT 0
EL EL 0 0
4/ i 0
I 0 .
Subtracting this from /C,,, we obtain the reduced 3 x 3 stiffness matrix for the square
frame with one pinned end.

El 6
=~T / 0.
3
/ 0,
Note that the middle column and row remain untouched.

6.6 ORTHOGONALITY OF EIGENVECTORS

The normal modes, or the eigenvectors of the system, can be shown to be
orthogonal with respect to the mass and stiffness matrices. By using the notation
for the ¿th eigenvector, the normal mode equation for the iih mode is
K(f)- = (6.6-1)
Premultiplying the /th equation by the transpose (f)J of mode j\ we obtain
cj>]Kct>^ = (6.6-2)
If next we start with the equation for the ;th mode and premultiplying by (f)J,
we obtain a similar equation with / and j interchanged:
(6.6-3)

Because K and M are symmetric matrices, the following relationships hold.

= 4,r 4>J (6.6-4)

Thus, subtracting Eq. (6.6-3) from Eq. (6.6-2), we obtain
(A, - = 0 (6.6-5)