Page 267 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 267
254 Computational Methods Chap. 8
Figure 8.10-1.
used in the transformation of coordinates to rotate the axes through an angle 0, as
illustrated in Fig. 8.10-1.
Matrix R is orthonormal because it satisfies the relationship
R^R = RR^ = I
In this case, there is only one ofl'-diagonal element, 2^ f^e eigenproblem is
solved in a single step. We have
cos 0 sin 0 «11 a,2 COS 0 -sin 0 A, 0
R]A,R, =
-sin 0 cos 0 a ,2 «22 sin 0 cos 0 0 ^ 2 .
Equating the two sides of this equation, we obtain
Aj = fljj cos^ 6 + 2«|2 sin 6 cos 0 + ¿^22 ^
A2 = sin^ 6 - 2aj2 sin 0 cos 6 + «22 0 (8.10-5)
0 = - (ajj - ^22) sin 0 cos 0 + aj2(cos^ 0 - sin^ 0)
From the last of Eqs (8.10-5), angle 0 must satisfy the relation
2^1-)
tan 20 (8.10-6)
^22
The two eigenvalues are then obtained from the two remaining equations or
directly from the diagonalized matrix. The eigenvectors corresponding to the two
eigenvalues are represented by the two columns of the rotation matrix R^, which
in this case is equal to P.
For the previous problem there was only one diagonal term a^j and no
iteration was necessary. For the more general case of the nth-order matrix, the
rotation matrix is a unit matrix with the rotation matrix superimposed to align with
the (/, j) off-diagonal element to be zeroed. For example, to eliminate the element