Page 413 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 413

Classical Methods   Chap. 12
                              400










                                                                 Q   Figure  12.8-1.
                                     n -1



                                                 'Jn
                                                   '^n

                          'n-\                         '  n
                                                       Jn(JJ^9n      Figure  12.8-2.
                                  In  the  development  so  far,  the  stations  were  numbered  in  increasing  order
                              from left to right with the transfer matrix also progressing to the right. The  arrow
                              under the equal sign indicates this direction of progression. In some problems, it is
                              convenient to proceed with the transfer matrix in  the opposite  direction,  in which
                              case we  need only to invert Eq.  (12.8-3).  We then obtain the relationship

                                                                       1_
                                                          1 -

                                                               K       K                 (12.8-4)
                                                                       1
                              The arrow now indicates that the transfer matrix progresses from right to left with
                              the  order  of  the  station  numbering  unchanged.  The  reader  should  verify  this
                              equation,  starting with the free-body development.


                       12.9  SYSTEMS WITH  DAMPING
                              When  damping is  included,  the form  of the  transfer matrix is not  altered, but the
                              mass  and  stiffness  elements become  complex quantities.  This  can be  easily shown
                              by writing  the  equations  for  the  nth  subsystem  shown  in  Fig.  12.9-1.  The  torque
                              equation  for disk  n  is
                                                        ,2l a  - jR _ JL  _   a
                              or
                                               (/o>c„  -     =  r„«  -   7 /             (12.9-1)
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