Page 387 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

374 Classical Methods Chap. 12

potential energies, an alternative equation for the fundamental frequency of the
beam is
, 2
dx
' - '1 0
---- ( 12.1-12)
¡y^ dm
Example 12.1-1
In applying this procedure to a simply supported beam of uniform cross section,
shown in Fig. 12.1-2, we assume the deflection to be represented by a sine wave as
follows:
. 1TX\ .
/
y = I Vqsm -j- I sm o)t

where yo is the maximum deflection at midspan. The second derivative then becomes
d^y / 77 . 7TX .

( j j >^oSin — sm (Ot
dx^
Substituting into Eq. (12.1-12), we obtain
El n -r / sm^—r -- dx
w ; -/q / , El
m j sm -j- dx

The fundamental frequency, therefore, is

In this case, the assumed curve happened to be the natural vibration eurve, and
the exact frequency is obtained by Rayleigh’s method. Any other eurve assumed for
the case can be considered to be the result of additional eonstraints, or stiffness,
which result in a constant greater than in the frequency equation.

Figure 12.1-2.

Example 12.1-2
If the distance between the ends of the beam of Fig. 12.1-2 is rigidly fixed, a tensile
stress a will be developed by the lateral deflection. Account for this additional strain
energy in the frequency equation.
Solution: Due to the lateral deflection, the length dx of the beam is increased by an
amount

[V’l + {dy/dxf - l] ii»: s f ( ^ ) dx