Page 374 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 11.5 Component-Mode Synthesis 361

Figure 11.5-1. Beam sections 1 and 2 with their coordinates.

We separate the beam into two sections, (D and (2), whose coordinates are
shown as Wj, x; W2, x; and U2, x. For part (D, we assume the deflection to be

Pi ( 1 1 . 5 - 1 )
Note that the two mode functions satisfy the geometric and force conditions at the
boundaries of section (D as follows:

w,(0) = 0 w^il) = P, + P2
tv',(0) = 0 W'',(/) = JP^ + J P 2
_ M(l) _ 2
w';(0) = = j^p^ (11 .5 -2)
El
£l i3 P2

Next consider part (2) with the origin of the coordinates W2, x at the free
end. The following functions satisfy the boundary conditions of beam section (2):
W2(x, t) = 4)2{x)p2{t) + (f)^{x)p^{t) + 4)^{x)p^{t) +

= lP3 + ( j ]pa + ( 7 ) P5 ( 1 1 . 5 - 3 )
U2(x,t) = 4>(,{x)P(^(t) + ■■■

(11.5-4)
= 1^6
where U2(x, t) is the displacement in the x-direction.