Page 395 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

382 Classical Methods Chap. 12

frequency of the system is determined as
2 2
^11^22 (3.515)" X 3.0 ( El = 2.41
iO11 + O ) (3.515)^ + 3.0 I A// (^)

This result can be compared to the frequency equation obtained by Rayleigh’s
method, which is
3 El
1 / 33 \ = 2 . 4 3 f ^ )
\Ml^)

Example 12.2-4
The natural frequency of a given airplane wing in torsion is 1600 cpm. What will be
the new torsional frequency if a 1000-lb fuel tank is hung at a position one-sixth of the
semispan from the center line of the airplane such that its moment of inertia about
the torsional axis is 1800 lb • in • s^? The torsional stiffness of the wing at this point is
60 X 10^ lb • in./rad.
Solution: The frequency of the tank attached to the weightless wing is

60 X lO'’
fl2 ~ 2tt 1800 = 29.1 cps = 1745 cpm
The new torsional frequency with the tank, from Eq. (a) of Example 12.2-1, then
becomes

/, = 1180 cpm
a (1600)" (1745)"

Example 12.2-5
The fundamental frequency of a uniform beam of mass M, simply supported as in
Fig. 12.2-2, is equal to El/MP . If a lumped mass is attached to the beam at
X = //3, determine the new fundamental frequency.
Solution: Starting with Eq. (b) of Example 12.2-1, we let iOjj be the fundamental
frequency of the uniform beam and o)j the new fundamental frequency with
attached to the beam. Multiplying through Eq. (b) by we have
1 = ( £ ! l)

Figure 12.2-2.