Page 396 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.2 Dunkerley’s Equation 383

-------- L _
1^11/ 1 + a22^{)0
The quantity a22 is the influenee coeffieient at x = 1/3 due to a unit load
applied at the same point. It can be found from the beam formula in Example 12.1-4
to be
8
6 X 81 £ /
Substituting o)jj = v'^El/Ml^ together with ^22^ obtain the convenient equation
1 1
1 + 1 + 1-6 at

6 X 81 M
Example 12.2-6
Determine the fundamental frequency of the three-story building shown in Fig.
12.2-3, where the foundation is capable of translation.
Solution: If a unit force is placed at each floor, the influence coefficients are
1

1
24EL

1
~ 24EE 24 El 2
1
24EI^ " 24EI2 24EI2

The Dunkerley equation then becomes
1 "*0 , 1 rrin
iO? Uo 24EI^ k. ^ 24EI 24EI2
1

^0 24E/i ^ 24E/2 24EE

Figure 12.2-3.