Page 332 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.5 Vibrations Involving Beam Elements 319

Coordinate reduction
Example 10.5-2
The solution of the preceding equation requires an eigenvalue-eigenvector computer
program. However, we consider a simpler problem of replacing the uniformly dis­
tributed mass by lumped masses at joints 2 and 3. The mass matrix is then a series of
Os except for elements /TI22 and This suggests rearranging the preceding
equation so that the displacement vector is in the rearranged order
(^'2\
^’3
62

This is simply accomplished by interchanging columns 2 and 3 and rows 2 and 3,
resulting in the following equation:
m 2 0 1 0 'V2 24 - 1 2 1 0 3/ io^
0 W3 I ¿\3 - 1 2 1 2 1 - 3 / - 3 / '■ 3 0
1 1
' +
0 1 0 1 e } 0 - 3 / ! 2/2 0.5/2 02

1
1 3/ - 3 / ! 0.5/2 /2
1 1
The equation is now in the form
' mu ! 0 ' ' V (10.5-1)

0 I 0 1w . .^21 1 ^ 22.

.
.
.
.
.
.
which can be written as
M,iK + a:,,k + = 0

K22O= 0
From the second equation, 8 can be expressed in terms of V:
8 =
Substituting into the first equation, we have
A/,|K+ ( a:,, - (10.5-2)
which in terms of the original quantities becomes
m 2 0 24 - 1 2 '
i'H + i — 1
0 m 3 \ihj W ' - 1 2 12.
- 1
0 3/ 2/2 0.5/2’ 0 -3/1
-3 / -3 /. 0.5/2 /2 3/ -3/J
The term
(10.5-3)
is the reduced stiffness, and its value when multiplied out is
SEI 96 -30 _ 48 F/ 16 -5
IP -30 12 IP -5 2