Page 412 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 412

Sec. 12.8   Transfer Matrices                                  399


                              for  the  symmetric  and  antisymmetric  modes  enables  sufficient  equations  for  the
                              solution.


                       12.8  TRANSFER  MATRICES

                              The  Holzer  and  the  Myklestad  methods  can  be  recast  in  terms  of  transfer
                              matrices.^  The transfer matrix defines the  geometric and  dynamic relationships of
                              the element between the two stations and allows the state vector for the force and
                              displacement to be  transferred from one  station  to the  next station.

                                  Torsional  system.  Signs  are  often  a  source  of  confusion  in  rotating
                              systems,  and  it  is  necessary to  clearly define  the  sense  of positive  quantities.  The
                              coordinate along the rotational axis is considered positive toward the right. If a cut
                              is  made  across  the  shaft,  the  face  with  the  outward  normal  toward  the  positive
                              coordinate  direction  is  called  the  positive  face.  Positive  torques  and  positive
                              angular  displacements  are  indicated  on  the  positive  face  by  arrows  pointing
                              positively according to the right-hand screw rule,  as shown  in  Fig.  12.8-1.
                                  With the stations numbered from left to right, the nih element is represented
                              by  the  massless  shaft  of torsional  stiffness   and  the  mass  of polar  moment  of
                              inertia  J^,  as shown in  Fig.  12.8-2.
                                  Separating  the  shaft  from  the  rotating  mass,  we  can  write  the  following
                              equations  and  express  them  in  matrix form.  Superscripts  L  and  R  represent  the
                              left and  right sides of the members.
                                  For the mass:

                                                =  e!;    I              1
                                                                                         ( 12.8-1)
                                                                       —(O^j

                                  For the  shaft:
                                                                        ■  i    e        ( 12.8-2)
                                                  TV  =  TJ^-1          0   1
                                  The  matrix pertaining to  the  mass  is  called  the  point  matrix  and  the  matrix
                              associated  with  the  shaft,  the  field matrix.  The  two  can  be  combined  to  establish
                              the transfer matrix for the  /ith element, which is
                                                  R   p
                                              (   \
                                               6         1
                                                                 K
                                                                                         (12.8-3)
                                                                  co^J
                                               T       -co^J
                                              \   / n
                                  ’^E. C. Pestel and F. A.  Leckie, ‘‘Matrix Methods  in  Elastomechanics,” (New York:  McGraw-Hill,
                              1963).
   407   408   409   410   411   412   413   414   415   416   417