Page 270 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.10 Jacobi Diagonalization 257

To complete the first sweep of all the off-diagonal terms, we next zero the term
la^ _ 2(0.2383) _
tan 26 = -0.7812
^11 - ^33 1.097 - 1.7071
26 = 37.996°
6 -18.998°
sin 6 -0.3255
eos 6 0.9455
0.9455 0 0.3255'
0 1 0
« 3 =
-0.3255 0 0.9455
1.0147 -0.0246 - 0.000
-0.0246 0.2134 0.0710 = ^3
0.000 0.0710 1.817
To further reduce the size of the off-diagonal terms, the procedure should be
repeated; however, we stop here and outline the procedure for determining the
eigenvalues and eigenvectors of the problem. The eigenvalues are given by the
diagonal elements of A and the eigenvectors of A are calculated from the products of
the rotation matriees 7?, as given by Eq. (8.10-9). These eigenvectors are of the
transformed equation in the y coordinates and must be converted to the eigenvectors
of the original equation in the x coordinate by Eq. (8.7-1). It should also be noted
that the eigenvalues are not always in the increasing order from 1 to n. In ^ 3,
Aj = (o]m/k is found in the middle of the diagonal.
X FROM A. COMPUTER VALUES
Aj = 0.213 Ai = 0.2094
A2 = 1.014 A2 = 1.000
A3 = 1.817 A, = 1.7905
It is seen here that even with one sweep of the off-diagonal terms, the results are in
fair agreement.
For the eigenvectors, we have
Y=R,R^R,
1 0 0 0.9530 0.3029 O' 0.9455 0 0.3255
0 0.7071 -0.7071 -0.3029 0.9530 0 0 1 0
0 0.7071 0.7071 0 0 1 -0.3255 0 0.9455
0.9011 0.3029 0.3102'
0.0276. 0.6739 -0.7383
-0.4327' 0.6739 0.5988
0.50 0 0 0.9011 0.3029 0.3102
X = U~^Y = 0 0.7071 0 0.0276 0.6739 -0.7383
0 0 1.00 -0.4327 0.6739 0.5988
0.4006 0.1515 0.1551'
0.0195 0.4765 --0.5221
-0.4327 0.6739 0.5988