Page 390 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.1 Rayleigh Method 377

and the bending moment beeomes

coirne
~T2T {V^ - Al^x + x^)
The maximum strain energy is found by substituting M{x) into i/max-
. 2 ,
1 / CO m e \ fl

^max = YËÎ dx
312
2T/ 144 135
The maximum kinetic energy is

X ‘ / ‘ 2 ’ /•' 4 , ‘ 2 ^

/max ^ 2 J ^ ^ ^ 6l> m J X dx = -^c (O^ m y
' f
•^0
By equating these results, we obtain
T T 4 W
= 3.531
ml^
which is very close to the exact result.
Lumped masses. The Rayleigh method can be used to determine the
fundamental frequency of a beam or shaft represented by a series of lumped
masses. As a first approximation, we assume a static deflection curve due to loads

M jg, A/2g, and so on, with corresponding deflections y ,, >^2^>"35 • • • • The
strain energy stored in the beam is determined from the work done by these loads,
and the maximum potential and kinetic energies become
^max - + • • ■) (12.1-17)
+ M ,y\ + M,y\ + • • • ) (12.1-18)
By equating the two, the frequency equation is established as
2 s^.Miyi
(12.1-19)

Example 12.1-4
Calculate the first approximation to the fundamental frequency of lateral vibration for
the system shown in Fig. 12.1-6.
225 kg 135 kg

2.5m L . 1.5m - j 1.5m
5.5m- Figure 12.1-6.