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
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