Page 392 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.2 Dunkerley’s Equation 379

12.2 DUNKERLEY’S EQUATION
The Rayleigh method, which gives the upper bound to the fundamental frequency,
can now be complemented by Dunkerley’s^ equation, which results in a lower
bound to the fundamental frequency. For the basis of the Dunkerley equation, we
examine the characteristic equation (8.3-2) formulated from the flexibility coeffi­
cients, which is
A) a,2^2 « 1 3 ^ 3

«22">2 - A) a 2^ rri = 0
V
«3l"î| a^2^2 ( ^ 3 3 ^ 3 - \
V 0)
Expanding this determinant, we obtain the third-degree equation in 1/co^
1 1
+ «22'^2 + ^ 33^ 3) + = 0 ( 12.2-1)

If the roots of this equation are 1/^2» 1/^3^ the previous equation can
be factored into the following form:
1 1 1 1 1 1 = 0
Ù ) (X), 0,2 coll (o,2 W3
or
0 ( 12.2-2)
2
\(0 ) \ (O (0^ I \ CO
As is well known in algebra, the coefficient of the second highest power is equal to
the sum of the roots of the characteristic equation. It is also equal to the sum of
the diagonal terms of matrix A ~ \ which is called the trace of the matrix (see
Appendix C):
. , ^ / 1
trace A = ¿^ \

These relationships are true for n greater than 3, and we can write for an
n-DOF system the following equation:
1 1 1
— + -h H---- ^ i22"^2 • (12.2-3)
CO^

^S. Dunkerley, “On the Whirling and Vibration of Shafts,” Phil. Trans. Roy. Soc., Vol. 185
(1895), pp. 269-360.