Page 376 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.5 Component-Mode Synthesis 363

-
The results computed for m^ and can now be arranged in the mass and

stiffness matrices partitioned as follows:
‘ 0.2000 0.1666 1 0 0 0 0
0.1666 0.1428 1 0 0 0 0
0 0 r 1.0000 0.5000 0.2000 ~1 0
[m] = ml (11.5-5)
0 0 0.5000 0.3333 0.1666 1 0
0 0 0.2000 0.1666 0.1111 1 0
_| _
0 0 0 0 1 1.0000
0
■ 4 6 1 0 0 0 0"
6 12 1 0 0 0 0
El 0 0 10 0 0 n 0

[/c] = (11.5-6)
P 0 0 1 0 0 0 10
0 0 0 28.8 10
Lo_ 4.
0 0 0 0 0 ! 0
where the upper left matrix refers to section (T) and the remainder to section .
At the junction between sections (D and we have the following con­
straint equations:
W^{l) -f «2(/) = 0 or Pi + P 2 +^6 = 0

w^{l) = 0 P3 + P4 + Ps = 0
,
- W’{1) = 0 2pi + 3p2 - P 4 - 4p5 = 0
EI[W[{1) + w"{/)] = 0 2 p| -l- 6p2 "1" 12p5 0
Arranged in matrix form, these are

{P\
1 1 0 0 1 Pi
0 0 1 1 0 Pi
2 3 0 - 4 0 P4 = 0 (11.5-7)
2 6 0 12 0 Pi
Pi
Because the total number of coordinates used is 6 and there are four
constraint equations, the number of generalized coordinates for the system is 2
(i.e., there are four superfluous coordinates corresponding to the four constraint
equations; see Sec. 7.1). We can thus choose any two of the coordinates to be the
generalized coordinates q. Let = q^ and = q^ be the generalized coordi­
nates and express p^,...,p^ in terms of q^ and ¿7^. This is accomplished in the
following steps.