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)