Page 377 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

364 Mode-Summation Procedures for Continuous Systems Chap. 11

Rearrange Eq. (11.5-7) by shifting columns 1 and 6 to the right side:
1 0 0 0 “ iP2\ ' - 1 -r
0 1 1 1 r -M - 0 0 (11.5-8)
3 0 - 1 - 4 1 - 2 0
6 0 0 12 - 2 0
In abbreviated notation, the preceding equation is
[^]{P2-5) = [Q]{(1i.6}
Premultiply by [i]^' to obtain
{Pl-i) =
Supply the identity and and write

{P\ -6} ^
This constraint equation is now in terms of the generalized coordinates q^ and ¿7^
as follows:
'iV 1 0
Pi - 1 - 1
Pi 2 4.50 = [C] (11.5-9)
‘ Pa -2.333 -5.0 \ Q . j •J6
Pi 0.333 0.50
W 0 1
Returning to the Lagrange equation for the system, which is
Fi
ml[m]{p] + = 0 (11.5-10)

substitute for [p] in terms of {q} from the constraint equation (11.5-9)
ml[m][C][q} + ^{ k] [C] [q } =
P
Premultiply by the transpose [C]':
E7,
[C]'[k][C]{q) = 0 (11.5-11)
P
Comparing Eqs. (11.5-10) and (11.5-11), we note that in Eq. (11.5-10), the
mass and stiffness matrices are 6 x 6 [see Eqs. (11.5-5) and (11.5-6)], whereas the
matrices [C]'[m][C] and [C]'[/c][C] in Eq. (11.5-11) are 2 x 2 . Thus, we have
reduced the size of the system from a 6 x 6 t o a 2 x 2 problem.
By letting [q] = -oj^iq], Eq. (11.5-11) is in the form
■ «11 «12 El bn b,2

—COml (11.5-12)
«21 «22 ^21 ^22 J ^6