Page 386 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.1 Rayleigh Method 373

explained on the basis that any deviation from the natural curve requires addi­
tional constraints, a condition that implies greater stiffness and higher frequency.
In general, the use of the static deflection curve of the elastic body results in a
fairly accurate value of the fundamental frequency. If greater accuracy is desired,
the approximate curve can be repeatedly improved.
In our previous discussion of the Rayleigh method, the potential energy was
determined by the work done by the static weights in the assumed deformation.
This work is, of course, stored in the flexible member as strain energy. For beams,
the elastic strain energy can be calculated in terms of its flexural rigidity EL
By letting M be the bending moment and d the slope of the elastic curve, the
strain energy stored in an infinitesimal beam element is
d U = j M d d (12.1-8)
Because the deflection in beams is generally small, the following geometric
relations are assumed to hold (see Fig. 12.1-1):
d^y
^ dx R dx dx^
In addition, we have, from the theory of beams, the flexure equation:

(12.1-9)
R ~ El
where R is the radius of curvature. By substituting for dd and 1/R, U can be
written as
, 2
^max c/x (12.1-10)
where the integration is carried out over the entire beam.
The kinetic energy is simply
1
•2 ^
T . = 7T y dm = 2 ^^ j d m ( 12.1-11)
/
where y is the assumed deflection curve. Thus, by equating the kinetic and

Figure 12.1-1.