Page 473 - Dynamics of Mechanical Systems

0593_C13_fm Page 454 Monday, May 6, 2002 3:21 PM

454 Dynamics of Mechanical Systems

Next, let ω be ω (that is, 2km ). By substituting into Eqs. (13.7.13), (13.7.14), and
2
(13.7.15), we obtain the equations:

A = 0 (13.7.24)
2
A =− A (13.7.25)
1 3

A = 0 (13.7.26)
2
These equations are also dependent (the first and third are the same). By using the first
two of these together with Eq. (13.7.19) and solving for A , A , and A , we have:
2
1
3
A = 22 , A = 0 , A = − 2 2 (13.7.27)
3
1
2
Finally, let ω be ω (that is, {(2 + 2 ) (k/m)} 1/2 ). Substituting into Eqs. (13.7.13), (13.7.14),
3
and (13.7.15), we obtain the equations:
2 A + A = 0 (13.7.28)
1 2

A + 2 A + A = 0 (13.7.29)
1 2 3

A + 2 A = 0 (13.7.30)
2 3
Observe that these equations are also dependent. (If we multiply the first and third by
2 and add them, we obtain a multiple of the second equation.) Using the first two of
these with Eq. (13.7.19) then produces:

A =− 12 , A = 2 2 , A = − 12 (13.7.31)
1 2 3
Equations (13.7.23), (13.7.27), and (13.7.26) represent the amplitude solutions corre-
sponding to the three frequencies of Eq. (13.7.18). Figures 13.7.4, 13.7.5, and 13.7.6 provide
a pictorial representation of these solutions depicting the movements of the particles of
the tube. We will discuss these solutions in greater detail in the following section.

n
3
3 x n 2
3
2 x
2
x n 1
1
FIGURE 13.7.3
Coordinates of particles in the fixed P P P T
horizontal tube. 1 2 3