Page 419 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

406 Classical Methods Chap. 12

By substituting Eq. (e) into Eq. (a), the transfer function of shaft A across the gears
becomes
1 0
(0
- 0 ) % / ( l - (O^J^/K^) 1
It is now possible to proceed along shaft A from \R to 3R in the usual manner.

12.12 TRANSFER MATRICES FOR BEAMS

The algebraic equations of Sec. 12.6 can be rearranged so that the four quantities
at station i + 1 are expressed in terms of the same four quantities at station i.
When such equations are presented in matrix form, they are known as transfer
matrices. In this section, we present a procedure for the formulation and assembly
of the matrix equation in terms of its boundary conditions.
Figure 12.12-1 shows the same ith section of the beam of Fig. 12.6-1 broken
down further into a point mass and a massless beam by cutting the beam just right
of the mass. We designate the quantities to the left and right of the mass by
superscripts L and R, respectively.
Considering, first, the massless beam section, the following equations can be
written:

^ i +\ v r
M h,

et,, = e t + — ] ( 12.12-1)
2EI

yt+ 1 = y t + 1( ^ j _ + +1 ( ¿ 7

Substituting for Vfi ^ and M/; j from the first two equations into the last two and

Figure 12.12-1. Beam sections for
transfer matrices.