Page 393 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

380 Classical Methods Chap. 12

The estimate to the fundamental frequency is made by recognizing that
o>2, ... are natural frequencies of higher modes and hence 1 /io\, 1 ... can
be neglected in the left side of Eq. (12.2-3). The term \/oy\ is consequently larger
than the true value, and therefore co^ is smaller than the exact value of the
fundamental frequency. Dunkerley’s estimate of the fundamental frequency is then
made from the equation

^ < I -f (12.2-4)
0),
Because the left side of the equation has the dimension of the reciprocal of the
frequency squared, each term on the right side must also be of the same
dimension. Each term on the right side must then be considered to be the
contribution to 1 /co] in the absence of all other masses, and, for convenience, we
let = 1/^*^?,, or
1 1 1 1
2 H------ ^ + • ' (12.2-5)
^11 ^ 2 2 K n
Thus, the right side becomes the sum of the effect of each mass acting in the
absence of all other masses.
Example 12.2-1
Dunkerley’s equation is useful for estimating the fundamental frequency of a struc­
ture undergoing vibration testing. Natural frequencies of structures are often de­
termined by attaching an eccentric mass exciter to the structure and noting the fre­
quencies corresponding to the maximum amplitude. The frequencies so measured
represent those of the structure plus exciter and can deviate considerably from the
natural frequencies of the structure itself when the mass of the exciter is a substantial
percentage of the total mass. In such cases, the fundamental frequency of the
structure by itself can be determined by the following equation:
1 1 1
(a)
2 2
^22
where co, = fundamental frequency of structure plus exciter

= fundamental frequency of the structure by itself
0)22 ^ natural frequency of exciter mounted on the structure in the absence of
other masses
It is sometimes convenient to express the equation in another form, for
instance,
1 1
(b)
Ù), CO]