Page 389 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

376 Classical Methods Chap. 12

/T? (x) y{x)dx

I
M \M-^dM
.dx:
Figure 12.1-4. Free-body diagram
V V-^dV of the beam element.
the moment at jc is found from the integral

M(x) = (12.1-15)

The strain energy of the beam is then found from
1 fiM{x)
dx (12.1-16)
El
which avoids any differentiation of the assumed deflection curve.
Example 12.1-3
Determine the fundamental frequency of the uniform cantilever beam shown in Fig.
12.1-5 using the simple curve y = cx^.

Figure 12.1-5.

Solution: If we use Eq. (12.1-12), we will find the result to be very much in error because
the previous curve does not satisfy the boundary conditions at the free end. By using
Eq. (12.1-12), we obtain

whereas the exact value is

Acceptable results using the given curve can be found by the procedure
outlined in the previous section.