Page 394 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.2 Dunkerley’s Equation 381

where nij is the mass of the eoncentrated weight or exciter, and <222 influence
coefficient of the structure at the point of attachment of the exciter.

Example 12.2-2
An airplane rudder tab showed a resonant frequency of 30 cps when vibrated by an
eccentric mass shaker weighing 1.5 lb. By attaching an additional weight of 1.5 lb to
the shaker, the resonant frequency was lowered to 24 cps. Determine the true natural
frequency of the tab.
Solution: The measured resonant frequencies are those due to the total mass of the tab
and shaker. Letting /jj be the true natural frequency of the tab and substituting into
Eq. (b) of Example 12.2-1, we obtain
1 1 1.5
386'*22
(277 X 30)"
1 1 3.0
386'^ 22
( 2 t7 X 24) ( 2 t7 / j|) "
By eliminating «22’ natural frequency is
/jj = 45.3 cps
The rigidity of stiffness of the tab at the point of attachment of the shaker can be
determined from l/« 22’ which from the same equations is found to be
J_ 1
k, = = 246 Ib/in.
d 22 0.00407
Example 12.2-3
Determine the fundamental frequency of a uniformly loaded cantilever beam with a
concentrated mass M at the end, equal to the mass of the uniform beam (see Fig.
12.2- 1).

M
M Figure 12.2-1.

Solution: The frequency equation for the uniformly loaded beam by itself is

For the concentrated mass by itself attached to a weightless cantilever beam, we have

. L - 3 . 0 0 ( | i )
By substituting into Dunkerley’s formula rearranged in the following form, the natural