Page 346 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.8 Generalized Force Proportional to Displacement 333

For il = 0, the natural frequencies for the single element analysis arc
= 3.53

ia. = 34.81
For comparison, the exact values for ii = 0 arc

w, = 3.515

ia. = 22.032
which indicates that the single-element analysis results in unacceptable accuracy for
the second mode. The eigenvectors for the single-element analysis, which are deflec­
tion and slope at the free end, cannot be compared to the more conventional
eigenvectors that display deflection along the beam.
Example 10.8-4 Two-Element Beam
If we divide the beam into two equal sections, then / is replaced by //2, and the
rotational force over the second clement must be changed to

^ dx
The generalized force is now
Q = mil}l ^ I ^ ^ drdx

= mil ‘PW',drdx\

The last integral in this expression is the same as that for the one-element beam,
except for / replaced by 1/2. The first integral now needs evaluating. In formulating
the equation of motion, this now requires changing all /’s in the matrices to 1/2.
We now suggest a different approach of leaving the length of each element
equal to /, so that the total length of the beam is 21. This results in great savings in
computation because the matrices for each clement will remain the same as that for
the one-element beam and all the /’s inside the matrices can remain as /, which can
be assigned as unity for the eigenvalue computation as before; i.e., we now solve the
problem shown in Fig. 10.8-3. After the eigenvalues arc determined, we let / in the
expression for the eigenvalues be replaced by y.

F
I (2) (3)

21- Figure 10.8-3.