Page 431 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

418 Classical Methods Chap. 12
12-44 Apply the matrix method to a cantilever beam of length / and mass m at the end, and
show that the natural frequency equation is directly obtained.
12-45 Apply the matrix method to a cantilever beam with two equal masses spaced equally
a distance /. Show that the boundary conditions of zero slope and deflection lead to
the equation
\m(x)^lK(^5 + \rrui)^l^K^
=
1 + U^Kmo)^
1 + + {\nuo"l^K)
~ 21 +
where K = I/EL Obtain the frequency equation from the foregoing relationship and
determine the two natural frequencies.
12-46 Using the matrix formulation, establish the boundary conditions for the symmetric
and antisymmetric bending modes for the system shown in Fig. P12-46. Plot the
boundary determinant against the frequency w to establish the natural frequencies,
and draw the first two mode shapes.

Figure P12-46.