Page 398 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 12.3 Rayleigh - Ritz Method 385

whereas for the longitudinal oscillation of slender rods,
= j d x and ^,7 = j d x

We now minimize co^ by differentiating it with respect to each of the
constants. For example, the derivative of co^ with respect to C, is

d(o^ ■max dC; ^ I = 0 (12.3-4)
9C, I 2
^ max
which is satisfied by
¿»i/max t^max '^Cax = 0
9C. 7m*ax

- (Ü ,
or because * - /.2
3T*
dU^ 2 ^ max _ Q (12.3-5)

dC, (?C,
The two terms in this equation are then
^Cax =
and
dC, = Lk.jCj
and so Eq. (12.3-5) becomes

o)^m,^) + C2{k,2 - (o^m,2) + ■■■ + C„(^,„ - = 0 (12.3-6)
With i varying from 1 to n, there will be n such equations, which can be arranged
in matrix form as

(^11 CO m. ,) (/C,2 - 0)2^12)
(^21 - <o^m2i)

{k„i - {k„„ co^m. C„
(12.3-7)
The determinant of this equation is an Ai-degree algebraic equation in and its
solution results in the n natural frequencies. The mode shape is also obtained by
solving for the C’s for each natural frequency and substituting into Eq. (12.3-2) for
the deflection.
Example 12.3-1
Figure 12.3-1 shows a wedge-shaped plate of constant thickness fixed into a rigid wall.
Determine the first two natural frequencies and mode shapes in longitudinal oscilla­
tion by using the Rayleigh-Ritz method.