Page 420 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.12 Transfer Matrices for Beams 407

arranging the results in matrix form, we obtain what is referred to as the field
matrix:
L p
i - V] 1 0 0 0 i - E l
M / 1 0 0 M
e
2 El El <
6 1 0 > ( 12.12-2)
e
y y
V J / + 1 6 El 2 El /
In this equation, a minus sign has been inserted for V in order to make the
elements of the field matrix all positive.
Next, consider the point mass for which the following equations can be
written:
yK = yL _

M!^ =
(12.12-3)
el^ = d!^

y," = yr
In matrix form, these equations become

- E l '' 1 0 0 mo)^ ( - V
M 1 0 1 0 0 1 M (12.12-4)

d I 0 0 1 0 1 ^

> 1. 0 0 0 1 1
'
which is known as the point matrix.
Substituting Eq. (12.12-4) into Eq. (12.12-2) and multiplying, we obtain the
assembled equation for the fth section:
R p
' - V 1 0 ma> ' - v \
M / 1 m(o^l M
,
e ■ e

6 ^ = mcD < e - (12.12-5)
2 El El 2 El
e e mo)^P
y 6E1 2 El 1 -f 6 El y
i+\ _
The square matrix in this equation is called the transfer matrix, because the
state vector at i is transferred to the state vector at / -(- 1 through this matrix. It is
evident then that it is possible to progress through the structure so that the state
vector at the far end is related to the state vector at the starting end by an